OF THE EUROPEAN MATHEMATICAL SOCIETY · Analysis and Stochastics of Growth Processes and Interface Models Peter Mörters, Roger Moser, Mathew Penrose, Hartmut Schwetlick and Johannes

NEWSLETTERO F T H E E U R O P E A N M A T H E M A T I C A L S O C I E T Y

FeatureCombinatorial Algebraic Topology

p. 13

History Mittag-Leffl er at Wernigerode

p. 23

Interview Mihailescu, Brezis

p. 27, 37

ERCOM Oberwolfach

p. 41

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EuropeanMathematicalSociety

June 2008Issue 68ISSN 1027-488X

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An Introduction to StochasticFiltering Theory Jie XiongStochastic filtering theory is a field that has seen arapid development in recent years. In this text, JieXiong introduces the reader to the basics ofStochastic Filtering Theory before covering the keyrecent advances. The text is written in a clear stylesuitable for graduates in mathematics andengineering with a background in basic probability.

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August 2008 | 504 ppHardback | 978-0-19-954511-7 | £75.00

Algebraic Models in GeometryYves Félix, John Oprea and Daniel TanréA text aimed at both geometers needing the tools ofrational homotopy theory to understand anddiscover new results concerning various geometricsubjects, and topologists who require greaterbreadth of knowledge about geometric applicationsof the algebra of homotopy theory.

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Analysis and Stochastics ofGrowth Processes and InterfaceModelsPeter Mörters, Roger Moser, Mathew Penrose, Hartmut Schwetlickand Johannes Zimmer

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Mathematics EmergingA Sourcebook 1540 - 1900

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Global Catastrophic RisksNick Bostrom, Milan M. Cirkovic andMartin J. ReesThis book focuses on Global Catastrophic Risksarising from natural catastrophes, nuclear war,terrorism, biological weapons, totalitarianism,advanced nanotechnology, artificial intelligence, andsocial collapse. Also addresses the most salientmethodological, ethical, and policy issues that arisein conjunction to the study of these threats.

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Invitation to DiscreteMathematics 2/eJirí Matoušek and Jaroslav Nešetril‘A far-from-traditional textbook and...a joy to read. Thetext is lucid and sprinkled with small jokes andbackground stories.’

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June 2008 | 456 ppPaperback | 978-0-19-857042-4 | £35.00 Hardback | 978-0-19-857043-1 | £75.00

American Mathematical Society

Contents

EMS Newsletter June 2008 1

Editors-in-Chief

Martin RaussenDepartment of Mathematical SciencesAalborg UniversityFredrik Bajers Vej 7GDK-9220 Aalborg Øst,Denmarke-mail: [email protected]

Vicente MuñozIMAFF – CSICC/Serrano, 113bisE-28006, Madrid, Spainvicente.munoz @imaff.cfmac.csic.es

Associate Editors

Vasile BerindeDepartment of Mathematicsand Computer ScienceUniversitatea de NordBaia MareFacultatea de StiinteStr. Victoriei, nr. 76430072, Baia Mare, Romaniae-mail: [email protected]

Krzysztof Ciesielski(Societies)Mathematics InstituteJagellonian UniversityReymonta 4PL-30-059, Kraków, Polande-mail: [email protected]

Robin WilsonDepartment of MathematicsThe Open UniversityMilton Keynes, MK7 6AA, UKe-mail: [email protected]

Copy Editor

Chris Nunn4 Rosehip WayLychpit Basingstoke RG24 8SW, UKe-mail: [email protected]

Editors

Giuseppe AnichiniDipartimento di Matematica Applicata “G. Sansone”Via S. Marta 3I-50139 Firenze, Italye-mail: giuseppe.anichini@unifi .it

Chris Budd(Applied Math./Applicationsof Math.)Department of Mathematical Sciences, University of BathBath BA2 7AY, UKe-mail: [email protected]

Mariolina Bartolini Bussi(Math. Education)Dip. Matematica – UniversitáVia G. Campi 213/bI-41100 Modena, Italye-mail: [email protected]

Ana Bela CruzeiroDepartamento de MatemáticaInstituto Superior TécnicoAv. Rovisco Pais1049-001 Lisboa, Portugale-mail: [email protected]

Ivan Netuka(Recent Books)Mathematical InstituteCharles UniversitySokolovská 83186 75 Praha 8Czech Republice-mail: [email protected]

Mădălina Păcurar(Conferences)Department of Statistics, Forecast and MathematicsBabes, -Bolyai UniversityT. Mihaili St. 58-60400591 Cluj-Napoca, [email protected]; [email protected]

Frédéric PaugamInstitut de Mathématiques de Jussieu 175, rue de ChevaleretF-75013 Paris, France e-mail: [email protected]

Ulf PerssonMatematiska VetenskaperChalmers tekniska högskolaS-412 96 Göteborg, Swedene-mail: [email protected]

Walter Purkert(History of Mathematics)Hausdorff-EditionMathematisches InstitutUniversität BonnBeringstrasse 1D-53115 Bonn, Germanye-mail: [email protected]

Themistocles M. Rassias(Problem Corner)Department of MathematicsNational Technical University of AthensZografou CampusGR-15780 Athens, Greecee-mail: [email protected].

Vladimír Souček(Recent Books)Mathematical InstituteCharles UniversitySokolovská 83186 75 Praha 8Czech Republice-mail: [email protected]

Editorial Team EuropeanMathematicalSocietyNewsletter No. 68, June 2008

EMS Calendar ............................................................................................................................. 2

Editorial ................................................................................................................................................. 3

Meetings in Copenhagen ............................................................................................ 5

Petition ................................................................................................................................................... 9

Abel Prize 2008 to Thompson and Tits .................................................. 10

Jahr der Mathematik 2008 – G. Ziegler ............................................................ 11

Combinatorial Algebraic Topology – D. Kozlov ................................13

On Platonism – R. Hersh ............................................................................................. 17

Mathematical Platonism and its Opposites – B. Mazur ....... 19

The meeting at Wernigerode – A. Stubhaug ................................... 23

Interview with P. Mihailescu .................................................................................. 27

Personal Column .................................................................................................................... 35

JEMS – an interview with H. Brezis .............................................................. 37

100th anniversary of ICMI – M. Bartolini Bussi ............................... 39

ERCOM: Oberwolfach .................................................................................................... 41

Book Review: Mittag-Leffl er biography – A. Jensen ............ 44

Forthcoming Conferences .......................................................................................... 46

Recent Books .............................................................................................................................. 51

The views expressed in this Newsletter are those of the authors and do not necessarily represent those of the EMS or the Editorial Team.

ISSN 1027-488X© 2008 European Mathematical SocietyPublished by the EMS Publishing HouseETH-Zentrum FLI C4CH-8092 Zürich, Switzerland.homepage: www.ems-ph.org

For advertisements contact: [email protected]

EMS News

2 EMS Newsletter June 2008

President

Prof. Ari Laptev(2007–10)Department of MathematicsSouth Kensington Campus, Imperial College LondonSW7 2AZ London, UKe-mail:[email protected]

andDepartment of Mathematics Royal Institute of Technology SE-100 44 Stockholm, Swedene-mail: [email protected]

Vice-Presidents

Prof. Pavel Exner(2005–08)Department of Theoretical Physics, NPIAcademy of Sciences25068 Rez – PragueCzech Republice-mail: [email protected]

Prof. Helge Holden(2007–10)Department of Mathematical SciencesNorwegian University of Science and TechnologyAlfred Getz vei 1NO-7491 Trondheim, Norwaye-mail: [email protected]

Secretary

Dr. Stephen Huggett(2007–10)School of Mathematics and Statistics University of Plymouth Plymouth PL4 8AA, UK e-mail: [email protected]

Treasurer

Prof. Jouko Väänänen(2007–10)Department of Mathematics and Statistics Gustaf Hällströmin katu 2b FIN-00014 University of Helsinki Finland e-mail: [email protected]

andInstitute for Logic, Language and ComputationUniversity of AmsterdamPlantage Muidergracht 241018 TV AmsterdamThe Netherlands e-mail: [email protected]

Ordinary Members

Prof. Victor Buchstaber(2005–08)Steklov Mathematical InstituteRussian Academy of SciencesGubkina St. 8Moscow 119991, Russiae-mail: [email protected] and [email protected]

Prof. Olga Gil-Medrano (2005–08)Department de Geometria i TopologiaFac. MatematiquesUniversitat de ValenciaAvda. Vte. Andres Estelles, 1E-46100 Burjassot, Valencia Spaine-mail: [email protected]

Prof. Mireille Martin-Deschamps (2007–10)Département de mathématiques Bâtiment Fermat 45, avenue des Etats-Unis F-78030 Versailles Cedex, France e-mail: [email protected]

Prof. Carlo Sbordone(2005–08)Dipartmento de Matematica “R. Caccioppoli”Università di Napoli “Federico II”Via Cintia80126 Napoli, Italye-mail: [email protected]

Prof. Klaus Schmidt(2005–08)Mathematics InstituteUniversity of ViennaNordbergstrasse 15A-1090 Vienna, Austriae-mail: [email protected]

EMS Secretariat

Ms. Riitta Ulmanen Department of Mathematics and StatisticsP.O. Box 68 (Gustaf Hällströmin katu 2b)FI-00014 University of Helsinki FinlandTel: (+358)-9-191 51507Fax: (+358)-9-191 51400Telex: 124690e-mail: ems-offi [email protected] Web site: http://www.emis.de

EMS Executive Committee EMS Calendar

2008

30 June–4 JulyThe European Consortium For Mathematics In Industry (ECMI), University College London (UK). www.ecmi2008.org

11–12 JulyEMS Executive Committee Meeting, Utrecht (The Netherlands) Stephen Huggett: [email protected]

12–13 JulyEMS Council Meeting, Utrecht (The Netherlands)Stephen Huggett: [email protected]; Riitta Ulmanen: ems-offi [email protected] http://www.math.ntnu.no/ems/council08/

13 JulyJoint EWM/EMS Workshop, Amsterdam (The Netherlands)http://womenandmath.wordpress.com/joint-ewmems-worskhop-amsterdam-july-13th-2008/

14–18 July5th European Mathematical Congress, Amsterdam (The Netherlands). http://www.5ecm.nl

18–22 JulyEuropean Open Forum ESOF 2008, session Can Mathematics help Medicine? Barcelona (Spain) http://www.esof2008.org

1 AugustDeadline for submission of material for the September issue of the EMS NewsletterVicente Muñoz: [email protected]

3–9 AugustJunior Mathematical Congress 8, Jena (Germany)www.jmc2008.org

16–31 AugustEMS-SMI Summer School at Cortona (Italy)Mathematical and numerical methods for the cardiovascular system [email protected]

8–19 SeptemberEMS Summer School at Montecatini (Italy)Mathematical models in the manufacturing of glass, polymers and textiles web.math.unifi .it/users/cime/

28 September – 8 OctoberEMS Summer School at Będlewo (Poland)Risk theory and related topicswww.impan.gov.pl/EMSsummerSchool/

7–9 NovemberEMS Executive Committee Meeting, Valencia (Spain) Stephen Huggett: [email protected]

2009

5–8 FebruaryEuropean Student Conference in Mathematics EUROMATH 2009, Cypruswww.euromath.org

Editorial


Dear Newsletter readers,

I have had the privilege to be in charge of the Newsletter of the European Math-ematical Society for almost fi ve years, since the autumn of 2003. Time has now come for me to hand over to my succes-

sor Vicente Muñoz from Madrid, Spain. He and I are jointly producing this issue and the next and he will take over responsibility after that. I plan to stay on the Newsletter’s editorial board and do bits of work here and then. Although I have been very fond of my job, of encouraging and fi nding interesting articles and as-sembling them in the Newsletter, I must admit that I am now looking forward to a period with more time for my primary occupation of mathematical research and teaching.

The production of a newsletter like this one is a col-laborative task. It could not have been done without a helpful and encouraging editorial board. You can fi nd the list of editors on page 2, and I have to thank all mem-bers for their contributions, whether they were produced from their own hands or by their intervention. Some of the editors have been on board much longer than I have and I hope that many of them will continue their good work in the future. This is not to say that the Newslet-ter could not use “fresh blood”, and I would like to en-courage volunteers to contact Vicente Muñoz who will compose a new editorial board at the beginning of next year.

Since 2005, the Newsletter has appeared under the EMS Publishing House – in a new format, both in print and electronically. Cooperation with director Thomas Hintermann has been very friendly and effi cient. I am aware of the fact that the money the EMS pays for the production of the Newsletter just pays the hard costs and I am therefore particularly grateful for the support from the publishing house. By the way, have a look at the impressive development of the publishing house both in terms of journals and of book publications at www.ems-ph.org! Cooperation with our Swiss layout ex-perts, Sylvia and Micha Lotrovsky, has been swift and friendly; likewise the handling of technical and thus TeXnical articles in the hands of Christoph Eyrich in Berlin. Thanks to all of them!

The main idea with this Newsletter is to give the per-sonal members of the EMS immediate “value” for their membership dues. The concept of the Newsletter has de-veloped over the years; nowadays, it contains one or two feature articles usually surveying a newer mathematical development, one article of a mainly historical character, at least one interview with a well-known mathematician, and a presentation of a national mathematical society

or of a mathematical research centre within Europe, and quite often also an article on news from Mathematical Education. Moreover, there are columns with news from the EMS, other interesting (mainly European) math-ematical news, a featured book review, a conference list-ing carefully produced in Romania, the section “Recent Books” from the hands of our Czech colleagues, and fi -nally (biannually) “Solved and Unsolved Problems” with comments and solutions, and a Personal Column.

It has not always been easy for this editor to fi nd vol-unteers willing to compose interesting feature articles. I am therefore extremely grateful for the support from editors of several membership journals of national math-ematical societies who have allowed me to “reuse” some of their articles and thus to make them visible to more mathematicians throughout Europe. Particular thanks go to the French gazette des mathématiciens, the German Mitteilungen der DMV and (my personal favourite) the Dutch Nieuw Archief voor Wiskunde.

It has been my privilege as editor to be invited, two or three times per year, to the weekend meetings of the ex-ecutive committee of the EMS. These take place in vari-ous European cities, usually by invitation of a national mathematical society. The meetings are pleasant because the participants work hard together for a common goal: a better infrastructure and a higher recognition in politics and in the public for European mathematics and mathe-maticians. Even more could certainly be done if the EMS had more active support (bottom-up) from its members – and from more members!

I do not know whether a leaving editor has the right to pronounce a wish. Anyway, I would like to address three wishes to the readers of this Newsletter issue – which this time is distributed in a higher circulation due to the European Congress in Amsterdam and to a campaign directed to all European departments of mathematics:

- Please inform the new editor-in-chief or one of the other editors about articles that you fi nd interesting for mathematicians at large, be they from your own or from somebody else’s pen. Every (email) alert will be gratefully acknowledged.

- The EMS would be able to work more effi ciently if it had more personal members – this would also in-crease the circulation of the Newsletter. Could you not try to convince one or more of your offi ce neigh-bours? Please have a look at the new EMS web page http://www.euro-math-soc.eu (still under construction). This page already offers many services to the commu-nity, e.g. announcements of jobs and conferences in ad-dition to news. Under Membership you will fi nd many more reasons to join the EMS; in addition, you can now become an EMS member online.

- If you fi nd this Newsletter worth reading, why not ask your department or your library for a subscription? Refer to http://www.ems-ph.org/journals/subscription.php?jrn=news .

I am grateful for your support in these matters.

EditorialMartin Raussen (Aalborg, Denmark)

EMS News


Following the tradition of joint mathematical weekends that was launched in Lisbon (2003) and continued in Prague (2004), Barcelona (2005) and Nantes (2006), the Danish Mathematical Society (DMF) organised a mathematical weekend in Copenhagen earlier this year starting on Friday 29 February and lasting until Sunday 2 March. More than 160 participants, many of them from abroad, listened to more than 50 lectures, making this the biggest mathematical event held in Denmark for years. To start with, the mayor of Copenhagen had invited the participants for a reception buffet at the city’s impressive town hall.

The meeting had been planned by an executive com-mittee consisting of four mathematicians from the Copen-hagen area, among them DMF president Søren Eilers, and moreover EMS vice-president Helge Holden. Time was provided for four plenary talks (by Xavier Buff, Toulouse; Nigel Higson, Pennsylvania State; Frank Merle, Cergy-Pontoise; and Stefan Schwede, Bonn) and six sessions:

- Algebraic topology (organizers: Jesper Grodal, Ib Madsen)

- Coding theory (Olav Geil, Tom Høholdt)- Non-commutative geometry/operator algebra (Ryszard

Nest, Mikael Rørdam)- Dynamical systems (Carsten Lunde Petersen, Jörg

Schmeling)- Algebra and representation theory (Jørn Børling

Olsson , Henning Haahr Andersen)- Partial differential equations (Gerd Grubb, Helge

Holden)

A detailed program can be found at http://www.math.ku.dk/english/research/conferences/emsweekend/.

Support for the meeting had been granted by the Danish science research council, the Danish and the Eu-ropean Mathematical Societies and the Department of Mathematics at Copenhagen University. A very enthu-siastic group of young mathematicians from this depart-ment must be thanked for their effi cient help with all the practical aspects of the meeting. For example, session talks were synchronized in order to allow participants to switch at the sound of a horn, which was impossible to miss by even the most enthusiastic speaker!

In summary, this was a very nice meeting that was useful for the whole audience, which ranged widely, not only with respect to their mathematical interests but also to their age; the attendees included not only many young PhD students but also several emeritus professors.

The mathematical weekend has established itself as a good framework for a general medium sized mathemat-ics conference. Who is going to organise the next one?

Concentrated audience

Group photo at the end of the conference

Joint mathematical weekend in CopenhagenMartin Raussen (Aalborg, Denmark)

EMS News


Arthur Besse donates royalties to EMS-CommitteeMireille Martin-Deschamps (Versailles Saint-Quentin, France)

The Friends of Arthur Besse are a group of mathema-ticians who work in the fi eld of Riemannian geometry. Here is their story. In 1975, Marcel Berger convinced his students to organise a workshop about one of his fa-vourite problems: understanding manifolds all of whose geodesics are closed. The workshop took place in Besse-en-Chandesse, a very pleasant village in the centre of France, and it turned out to be so successful that the de-cision was made to write a book about the topic. Arthur Besse was born.

The experience was so pleasant and enjoyable that Arthur did not stop there and settled down to write an-other book. This second book “Einstein manifolds” was eventually published in 1987.

Years have passed since then. Arthur’s friends have scattered to various places. As they say “for Arthur him-self, who never aimed to immortality, it may be time for retirement”.

Nevertheless, a new version of Einstein Manifolds has recently been published and the royalties have been of-fered to the EMS with the suggestion that this money could be used to help young people from developing countries. Of course this offer was gratefully accepted by the Executive Committee of the EMS, who decided to put this money at the disposal of the Developing Coun-tries Committee.

To give a rough estimate, in the long run these duties could provide 9000 euros.

EMS News


EMS Executive Committee meetingCopenhagen, Denmark, 3 March 2008Vasile Berinde, EMS Publicity Officer

Organized as a continuation of the successful Joint Math-ematical Weekend EMS – Danish Mathematical Society (February 29th–March 2nd), the fi rst EC meeting of 2008 was held at the Institute for Mathematical Sciences, Uni-versity of Copenhagen, on Monday, 3rd of March. This was an unusual working day, for, by tradition, the EC usually only meets at week-ends.

Except for a last minute cancellation by Carlo Sbordone, all members of the Executive Committee were there: Ari Laptev (President, in the Chair), Pavel Exner and Helge Holden (Vice-Presidents), Stephen Huggett (Secretary), Jouko Vaananen (Treasurer), Olga Gil-Medrano, Mireille Martin-Deschamps, Victor Buchstaber, and Klaus Schmidt (members), together with Riitta Ulmanen, our quiet and effi cient executive secretary, Martin Raussen, the current Editor-in-Chief of the Newsletter, his successor (starting with the September issue), Vicente Muñoz, who thus made his début at an EC meeting, Mario Primicerio (represent-ing applied mathematics within EC), and myself.

After a healthy and refreshing half hour walk trip from Cabinn Scandinavia to the meeting venue, people felt ready to run over the agenda of the day. Our Presi-dent, imperturbably relaxed, chaired an extremely dense marathon EC meeting, that started at 9.00 a.m. and lasted until almost 6.00 p.m., only broken for coffee and a quick lunch.

From the business of what was a very pleasant and effi cient meeting, I’ll mention just a few matters:

- President’s report (the effect of his letter to the Presi-dents of national mathematical societies; his very suc-cessful visits to Strasbourg and Brussels; his valuable visit to Serbia, the ISE matters)

- Treasurer’s report (on the fi nancial year 2007)- Publicity Ofi cer’s report (the database of European

departments of mathematics, the EMS poster)- Membership matters (new applications from Montene-

gro, Serbia, and Turkey; the outline of a fee structure; the new individual members formally accepted by the EC)

- EMS Web Site (the EC thanked Helge Holden by ac-claim for the progress with the web site that he dem-onstrated by taking the EC through the various pages; various suggestions for improvement were also made; our main web site will be freely maintained by Univer-sity of Bremen, while the fees payment pages would be hosted in Helsinki, and linked to the membership database)

- 5th European Congress of Mathematics (the EMS would have a booth at the 5ECM, shared with the EMS publishing house)

- 6th European Congress of Mathematics (all three bid-ders – Krakow, Prague and Vienna – would be able to see all three revised bids; at the Utrecht Council, each bid would be given 15 minutes for their presentation, plus some time for questions)

- Reports by standing committees- Publishing (the EC confi rmed the next Editor of the

Newsletter, Vicente Muñoz, following an informal meeting the previous day, where he gave a report on his editorial views)

- Closing matters (The President expressed the gratitude of all present to Søren Eilers and to Martin Raussen for the excellent arrangements which they had made for our meeting in Copenhagen).

The next EC meeting, in Utrecht, preceding the EMS Council, would start at 14.00 on the 11th of July, and would continue from 09.00 to 11.00 on the 12th, the day the EMS Council meeting starts. The last EC meeting of 2008 will be in Valencia, 7th–9th of November.

Copenhagen town hall

News


What happened to our friend and colleague Ibni Oumar Mahamat Saleh from Chad?Aline Bonami (Orléans, France) and Marie-Françoise Roy (Rennes, France)

The French learned socie-ties of mathematics: SMF (Société Mathématique de France), SMAI (Société de Mathématiques Appli-quées et Industrielles) and SFdS (Société Française de Statistiques) launched the following petition on 10 March 2008; it can be found at http://smf.emath.fr/PetitionSaleh/.

Mr. President of the Republic of Chad, Mr. President of the French Republic

We want to know the truth concerning our colleague, the mathematician Ibni Oumar Mahamet Saleh, a Chadian politician and former minister. He was abducted from his home on 3 February 2008 and we have had no news from him since then.

Ibni Oumar Mahamat Saleh is, beyond his political activities, an active member of the mathematical com-munity.

He completed his mathematical education at the Uni-versity of Orléans, where he defended his PhD thesis in 1978. He was appointed as a professor at the University of N’Djamena in 1985. His numerous positions at the university included:

- Chair of the Department of Mathematics (1985), - Director of the Center of Scientifi c Research (1986), - Rectorship (1990–1991).

In spite of his heavy administrative and ministerial du-ties, Ibni Oumar Mahamet Saleh always succeeded in maintaining a high level of teaching standards. In or-der to improve the scientifi c level of teachers at the

University of N’Djamena, he negotiated in 1991 a col-laboration between the University of Orléans and the University of N’Djamena, in association with INSA in Lyon and the University of Avignon. This agreement is still in force and has been very successful. In addition to its goal of training, it has allowed Chadian teachers to establish stable and fruitful contacts with European and African Universities. Even when called to other respon-sibilities, Ibni Oumar Mahamet Saleh was instrumental in the success of the project. Several times he visited the department of mathematics in Orléans as part of this framework.

Since 3 February, contradictory information has circu-lated about Ibni Oumar Mahamat Saleh. The two other opponents, who were arrested the same day, have been released. One of them thinks that our colleague died, while his family thinks that he is still alive.

We want to know the truth. »The petition has received over 2450 signatures from

the mathematical community around the world, particu-larly from many mathematicians in Africa who knew Ibni as a colleague or as a professor.

The petition, the list of signatories, messages of infor-mation to the signatories and several documents about the case can be found at http://smf.emath.fr/Petition-Saleh/.

A blog of information has been set up by his fam-ily and friends at http://prisonniers-politiques.over-blog.com/.

So far we have received no answer from the Chadian and French authorities and we fear that we may not to see our friend and colleague again.

We invite you to sign the petition.

Aline Bonami [[email protected]] and Marie-Françoise Roy [[email protected]]

News


The Centre de Recerca Matemàtica (CRM, Bellaterra, Spain) organizes, jointly with the EMS, a scientifi c ses-sion at ESOF2008.

The title of the session is Can Mathematics help Medi-cine? It is part of the activities on the theme “Engineer-ing the Body”.

The programme is as follows:Monday, July 21, 8:30–10:00

Alfi o Quarteroni (Politecnico di Milano, Italy and Ecole Polytechnique Fedérale Lausanne, Switzerland)Mathematical models for the cardiovascular system

Dominique Chapelle (INRIA-Rocquencourt, France)Modeling and estimation of the cardiac electromechani-cal activity

Peter Deufl hard (Zuse Institute Berlin, Germany)The challenge of electrocardiology: model hierarchy and multiscale simulation.

Moreover, ESOF2008 hosts the following plenary talks:

Marcus Du Sautoy (University of Oxford, UK)Mathematics: creative art or useful science?

Eva Bayer-Flückiger (Ecole Polytechnique Fédérale Lausanne, Switzerland)The Science of Communication: number theory and coding

For more information, please consult http://www.esof2008.org .

SIMAI2008 International conference in Rome (Italy) organized in cooperation with SIAM

The Italian Society for Applied and Industrial Mathematics (SIMAI) will hold its 9th Con-

gress in Rome, Italy from September 15th to 19th . This international event takes place every two years. This time it is being organized in cooperation with SIAM. The hosting environment will be the beautiful down-town of Rome.

The main (invited) speakers will be Antonio Ambrosetti (SISSA, Trieste), Douglas N. Arnold (University of Minne-sota, USA), Nicola Bellomo (Politecnico di Torino), Gio-vanni Ciccotti (Università di Roma ‚‘La Sapienza’’), Nicho-las J. Higham (University of Manchester, UK), and Alfi o Quarteroni (Politecnico di Milano & EPFL Lausanne).

Besides, there will be a number of minisymposia (usu-ally several dozens), round tables, prizes to young math-ematicians, and interactions with industrial representa-tives. More information (in progress) can be found at the SIMAI site http://www.iac.rm.cnr.it/simai/simai2008/ .

Note that all associates to a member society of EMS are entitled to the reduced conference fee as the mem-bers of SIMAI.

European Student Conference in MathematicsEUROMATH–20095–8 February 2009, Cyprus; www.euromath.org

This is a conference organized by the Cyprus Mathematical Society in co-operation with the European Math-ematical Society, the Ministry of Edu-cation of Cyprus, the University of Cyprus and the Thales Foundation.

The conference will consist of several workshops/sympo-siums covering many themes. Suggestions for workshops, symposiums, sessions and exhibitions for the conference are welcome. More information can be found at www.eu-romath.org; alternatively, please contact the chair of the organizing committee at [email protected]

The themes of interest are: applications of mathe-matics, mathematics and sciences, mathematics and life, mathematics and technology, mathematics and social sciences, mathematics and space, fractals and geometry, mathematics and economy, mathematics and literature, mathematics and music, mathematics and law, mathemat-ics and statistics, the history of mathematics, mathematics and society, mathematics and Europe, mathematics and philosophy, mathematics and computer science, and fa-mous numbers.

Students aged between 12 and 18 from any European or international school who may be interested in attend-ing should send a message with their full address, fax and email to the organizing committee. Registration should be completed by 30 November 2008. The deadline for submission of abstracts is 17 October 2008. Presented papers will be reviewed and invited for publication in the proceedings of the conference, which will be published after the event.

The registration fee is 75 euros for students and 150 euros for teachers.

News


Abel Prize 2008The Norwegian Aca-demy of Science and Letters has decided to award the Abel Prize for 2008 to John Griggs Thompson (Gradu-ate Research Pro-fessor, University of Florida) and Jacques Tits (Professor Eme-ritus, Collège de France, Paris)

“for their profound achievements in algebra and in par-ticular for shaping modern group theory”.

Modern algebra grew out of two ancient traditions in mathematics, the art of solving equations, and the use of symmetry as for example in the patterns of the tiles of the Alhambra. The two came together in late eighteenth century, when it was fi rst conceived that the key to un-derstanding even the simplest equations lies in the sym-metries of their solutions. This vision was brilliantly real-ised by two young mathematicians, Niels Henrik Abel and Evariste Galois, in the early nineteenth century. Eventu-ally it led to the notion of a group, the most powerful way to capture the idea of symmetry. In the twentieth century, the group theoretical approach was a crucial ingredient in the development of modern physics, from the under-standing of crystalline symmetries to the formulation of models for fundamental particles and forces.

In mathematics, the idea of a group proved enor-mously fertile. Groups have striking properties that unite many phenomena in different areas. The most important groups are fi nite groups, arising for example in the study of permutations, and linear groups, which are made up of symmetries that preserve an underlying geometry. The

work of the two laureates has been complementary: John Thompson concentrated on fi nite groups, while Jacques Tits worked predominantly with linear groups.

Thompson revolutionised the theory of fi nite groups by proving extraordinarily deep theorems that laid the foundation for the complete classifi cation of fi nite simple groups, one of the greatest achievements of twentieth cen-tury mathematics. Simple groups are atoms from which all fi nite groups are built. In a major breakthrough, Feit and Thompson proved that every non-elementary simple group has an even number of elements. Later Thompson extended this result to establish a classifi cation of an im-portant kind of fi nite simple group called an N-group. At this point, the classifi cation project came within reach and was carried to completion by others. Its almost incredible conclusion is that all fi nite simple groups belong to certain standard families, except for 26 sporadic groups. Thomp-son and his students played a major role in understand-ing the fascinating properties of these sporadic groups, including the largest, the so-called Monster.

Tits created a new and highly infl uential vision of groups as geometric objects. He introduced what is now known as a Tits building, which encodes in geometric terms the alge-braic structure of linear groups. The theory of buildings is a central unifying principle with an amazing range of ap-plications, for example to the classifi cation of algebraic and Lie groups as well as fi nite simple groups, to Kac-Moody groups (used by theoretical physicists), to combinatorial geometry (used in computer science), and to the study of rigidity phenomena in negatively curved spaces. Tits’s geometric approach was essential in the study and realisa-tion of the sporadic groups, including the Monster. He also established the celebrated “Tits alternative”: every fi nitely generated linear group is either virtually solvable or con-tains a copy of the free group on two generators. This result has inspired numerous variations and applications.

The achievements of John Thompson and of Jacques Tits are of extraordinary depth and infl uence. They com-plement each other and together form the backbone of modern group theory.

This citation is taken from www.abelprisen.no

Visit the Abel web site: www.abelprisen.no/en/abel.Under the auspices of the Abel prize and with EMS vice-president Helge Holden (NTNU, Trondheim) as the driv-ing force, a comprehensive web site collecting together the available historical material about Niels Henrik Abel (1802–1829) has been established. It contains:

- The only known painted portrait of Abel.- A personal biography (by Arild Stubhaug) and a scientifi c biography (by Christian Houzel).- Abel’s collected works and related material, scanned and available in pdf format.- Even several of Abel’s original handwritten manuscripts.

Moreover, the web site contains biographical and mathematical literature on Niels Henrik Abel and his work (most of it scanned) and photos of Abel memorabilia (monuments, stamps, banknotes, medals and so on).

Have a look and enjoy!

John Griggs Thompson, Jacques Tits (Photo: University of Florida and Jean-François Dars/CNRS Images)

News


Mathematics Year 2008 in GermanyGünter M. Ziegler (Berlin)

2008 was offi cially declared a “Mathematics Year” in Germany. This created an unprecedented opportunity to work on the public’s view of the subject. Although Mathematics Year 2008 is primarily “a German affair”, I believe that a number of the lessons we have learnt in preparing the Year and in promoting it to the media may be of interest to the international readership of the EMS newsletter.

Since 2000, which was a “Year of Physics” in Germany, the German Federal Ministry of Science and Education has dedicated each year to one particular science. Top-ics have been the natural sciences like biology, chemistry, geology and computer science, and also humanities in 2007. The Science Years in Germany have now acquired a number of well-tested and successful components:

- Big events, among them the opening and closing gala and the week-long “Science Summer”.

- Exhibitions, including a large one on the “Science Ship” that travels on the Rhine, Danube and Elbe rivers all summer.

- A major media and public relations campaign.

A fourth component is new for 2008; much more than in previous science years we are working to reach the schools (teachers, parents and thus students).

The motto for Mathematics Year 2008 is “Mathemat-ics. Everything that counts!”. The posters and activities for the schools declare: “You’re better at math than you think!”.

Four partners carry Mathematics Year 2008: besides the Federal Ministry of Science and the “Science in Dia-logue” agency (which represents the German Science Foundation and the major research organizations such as the Max Planck Society), there are the Deutsche Tele-kom Foundation and the German Mathematical Society (DMV). I am acting as the DMV president; it is impor-tant to have an offi cial function when communicating with politicians. We have a budget of roughly 7.5 million euro for the year; that sounds like a lot of money but the money is soon gone when you get into professional public relations, organizing big events, etc.

A professional event and advertising agency in Berlin has designed the look of the Year (logo, print and web design), organized the major events and run the editorial (and campaign) offi ce. However, in contrast to the approach of previous years, we insisted on having an additional “con-tents offi ce” for the media work. This is where we take advantage of expertise from the community, make sure it is represented in the Year, and make sure that there’s “math inside” the publications (and that the mathematics is correct). The hope is that when the Year is over, the mathematics content offi ce will keep running as an active platform for promoting mathematics to the public.

The four major partners that run the year have iden-tifi ed several aims that they would like to attain or that they intend to communicate. One is that mathematics is multi-faceted. It does include “learning to calculate” but also much more – mathematics is high-tech, it is art, it is puzzles, etc. Our main message for the year is: “There is lots to discover!”. People who think they don’t like math-ematics haven’t seen much of it. Therefore we try to show people sides of mathematics they have not seen yet – or show them aspects that they had not identifi ed as being connected to mathematics.

About one hundred large and small exhibitions all over Germany present different aspects of mathematics: mathematics institutes show historic objects and science centres have developed hands-on objects for children of different age groups (some exhibitions focus on numbers, i.e. where they come from, what they are good for, num-bers in nature, numbers in everyday life, lucky numbers and so on). The Mathematical Research Institute Ober-wolfach has developed and promoted an exposition “Im-aginary”, showing singularities of algebraic surfaces and

Opening session, from left to right: Klaus Kinkel, president of the German Telekom Foundation, Federal Minister of Education and Research Annette Schavan, DMV president Günter M. Ziegler, and Gerold Wefer, president of Science in Dialogue

News


other visually-striking geometric objects, mostly in three dimensions and partly rendered as colourful plots on plexiglass. With the relevant software – free to download – every visitor has the chance to create objects them-selves. A major German weekly newspaper (DIE ZEIT) has launched a competition for the most beautiful object created on its web site using the Imaginary software.

We frankly and actively admit that doing mathematics is diffi cult. Don’t try to say it’s all easy – it is not true and peo-ple will not believe you. We rather argue that mathematics is diffi cult but you need it. Since mathematics is diffi cult it is interesting for the brightest. We state that mathematics is everywhere – it’s just that one often doesn’t notice it. This has turned out to be an aspect that the media are very in-terested in. There are various columns in newspapers and magazines that focus on mathematics in everyday life. One point we stress is that every one of us uses mathematics daily without thinking: when comparing prices of goods, handing over the right amount of cash to a salesperson, knowing how long to park or to travel here and there, etc. The other point we make is that mathematics is hidden in many objects in daily use like iPods (compression tech-niques), automobiles (aerodynamics, navigation, optimiza-tion of electronics), communication, medicine, drug design and even sports (tactical manoeuvres, scoring systems).

We also think it is important to show that not only doing mathematics but also watching mathematics can be fun – for instance as part of fi lm plots. An active group of mathematicians connected to my colleague Konrad Polthier from Free University in Berlin have organized a Mathematics Film Festival with international help. They created a database of mathematics fi lms ranging from famous blockbusters to local low budget productions at universities. They negotiated contracts with fi lm distribu-tion companies under which some major movies have become available for free public viewing occasions. Some fi fty people and institutions in Germany have now regis-tered on the database and have consequently announced local fi lm festivals consisting of mathematics fi lms from this database.

Even if only a third of the year is over at the time of writing, there are several lessons that we have learned al-ready:

1. Don’t try to teach. There’s no hope that people will know more mathematics at the end of the year. If many people think of mathematics as something in-teresting at the end of the year, we will have been very successful.

2. Images, colours, graphics and photographs are impor-tant. Several mathematics calendars were produced for the year, with some great images, and they imme-diately sold out!

3. Faces are also important. A subject is “abstract” for the media as long as they don’t have people to talk to and individuals to write about. For all the press ma-terials, we are presenting and portraying mathemati-cians “to talk to”.

4. Talk to the press. Press releases are one thing but you also have to talk to the key editors about topics that

you can present especially for them. My experience is that they are interested!

5. Use professionals. For us, the year is an opportunity to get help and learn from an advertising agency but also from all the other major players. For example, the Deutsche Telekom Foundation has been funding mathematics education projects for years and they are also sponsoring new programs for the education and development of mathematics teachers.

6. Make it a community effort. We are working hard to get hundreds of people from all over Germany involved, inviting people to become “math makers” for the Year. This is the only way to have activities all over the country. A “top down” campaign cannot have a broad effect. 560 people had already been reg-istered as “math makers” by the end of April.

7. Use the opportunities. The physics community (rep-resented by the German Physics Society) profi ted a lot from the Year of Physics 2000 and used it to build infrastructure, enlarge their membership base and professionalize their web, print and media appear-ance. Mathematicians all over Germany are working hard to grab the opportunities.

This is my personal, preliminary collection of lessons learned. It might be revised in the course of the year. Still, many different people and groups and most of the media are stating that the Year of Mathematics in Germany is already a success. It is worth the effort and I would like to encourage mathematicians in Europe to start similar projects. If such opportunities arise for greater, national visibility for mathematics projects then grab them.

A huge campaign like the German Mathematics Year of course takes a lot of effort but it defi nitely seems to be worth that effort. We’ve made a major try and have got some major aspects right but we are also eager to learn in the process. Thus, your comments are very welcome!

Günter M. Ziegler [[email protected]] is a professor of mathematics at Technische Universität Ber-lin and the president of the German Mathematical Society (DMV). He acknowledges the kind assistance with this article of Thomas Vogt [[email protected]] from the Mathematics Year Contents Offi ce situated at Technische Universität Berlin.

Swimming science center with a math exhibition

Feature


Combinatorial Algebraic TopologyDmitry N. Kozlov (Bremen, Germany)

1 Introduction

Combinatorial Algebraic Topology is a contemporary mathe-

matical discipline which transcends several classical branches

of study. Discrete mathematics and algebraic topology should

perhaps be mentioned in the first batch, but further connec-

tions exist to such areas as algebra and category theory. Out-

side of classical mathematics there are important applications

as well – we mention here connections to theoretical com-

puter science, via complexity theory, and to vision recogni-

tion, via point cloud data and persistent homology.

The dramatis personae of Combinatorial Algebraic Topol-

ogy are cell complexes which are combinatorial both locallyand globally. Being combinatorial locally is a well-studied

feature in algebraic topology. It means that the cell attach-

ment maps can be described by discrete data rather than by

arbitrary continuous maps. Often one simply considers sim-

plicial complexes, but also more intricate constructions ex-

ist. Being combinatorial globally is a feature more intrinsic

to Combinatorial Algebraic Topology. Expressed succinctly,

it means that the cells are indexed by combinatorial objects

and that the cell attachments are encoded by some combina-

torial rules, or more generally by algorithms, applied to the

combinatorial objects which index the involved cells.

The main aspiration of the subject is then to understand

how the algebraic invariants of such combinatorial cell com-

plexes, e.g., their Betti numbers, depend on the discrete data.

At best one may be able to completely describe these alge-

braic structures in the language of combinatorics; either by

using existing gadgets, such as matroids, or by inventing new

ones. The mechanisms of computation themselves are some-

times purely combinatorial – such as clever machinations with

labellings, orderings, and the like. When these fail, adapta-

tions of central methods of computation used in algebraic

topology, such as spectral sequences, can be tried.

Rather than risking to confuse the reader with too many

general words, we have chosen to present our subject by means

of three examples. In the first one we show how discrete meth-

ods can be used in existing topological situations. In the sec-

ond one the tables are turned: Topological constructions are

used to gain knowledge about discrete objects. The last ex-

ample highlights connections between algebraic topology and

complexity theory.

2 Goresky-MacPherson formula inarrangement theory

Combinatorial Algebraic Topology can help to analyze topo-

logical spaces which are important elsewhere if the invariants

of the spaces which we would like to compute are completely

determined by the combinatorial data. When they are, we of-

ten end up having a meaningful and deep connection between

topology and combinatorics.

One representative case is determining some of the most

important invariants, i.e., the cohomology groups of certain

spaces related to arrangements. To start with recall that a sub-space arrangement is a collection A = {A1, . . . ,An} of finitely

many linear subspaces in kd , where k = R or k = C, such that

Ai �⊇ A j for i �= j. One of the spaces related to such an arrange-

ment is its complement MA := kd \⋃ni=1 Ai whose algebraic

invariants are of interest in various applications in such areas

as algebraic geometry and singularity theory.

A classical example is obtained by taking the special hy-

perplane arrangement Ad−1, called the braid arrangement,consisting of all hyperplanes xi = x j, for 1 ≤ i < j ≤ d. The

complement of the braid arrangement Ad−1 in Rd consists of

d! contractible pieces, indexed by all possible orderings of the

coordinates.

The combinatorial data which is standardly associated to

an arbitrary subspace arrangement A is the so-called inter-section lattice LA , which is the set{

KI =⋂i∈I

Ai, for I ⊆ {1, . . . ,n}}∪{kd}

with the order given by reversing inclusions: x ≤LAy if and

only if x ⊇ y. In particular, the minimal element of LA is kd ,

denoted 0, and the maximal element is⋂

KI∈A KI . For exam-

ple, in the case of the braid arrangement, the intersection lat-

tice is the so-called partition lattice Πd . This is the lattice

whose elements are indexed by set partitions of {1, . . . ,n},

with the partial order given by refinement.

It was discovered by Goresky and MacPherson, as a corol-

lary of their Stratified Morse theory, see [GoM88], that the co-

homology groups of the complement of a subspace arrange-

ment A are determined by the combinatorial data LA :

Theorem 2.1. (Goresky-MacPherson)

For an arbitrary subspace arrangement A we have

Hi(MA ) ∼=⊕

x∈L ≥0A

HcodimR(x)−i−2(∆(0,x)),

where ∆(0,x) denotes the nerve of the interval in LA viewedas a category.

The last words of Theorem 2.1 need a little explanation. As

defined, LA is a partially ordered set, hence so is the interval

(0,x). The latter consists of all elements y of LA , such that

0 < y < x, equipped with the partial order induced from the

one on LA . Any poset P can be seen as a category whose

elements are the elements of P, with a unique morphism from

x to y, for x,y ∈ P, whenever x ≥ y. The transitivity of the

partial order guarantees the existence of the composition and

the fulfilment of the associativity axiom.

There is a standard way to associate a topological space,

in fact, even a structure of a simplicial set, to any category,

which is called the nerve of that category. In the special case


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of a poset we get a simplicial complex, the so-called ordercomplex, whose vertices are the elements of the poset, and

whose simplices are all chains (fully ordered subsets) of P.

There are many combinatorial tools to determine cohomol-

ogy groups of an order complex of a poset. Thus, in particular,

Theorem 2.1 means that the cohomology groups of the com-

plements of specific, combinatorially defined arrangements

can often be found by purely discrete methods.

One can turn a real subspace arrangement A into a com-

plex one A C by taking the linear subspaces in Cd defined by

the same equations as those in A . Certainly, this new com-

plexified arrangement has the same intersection lattice. We

therefore obtain a curious corollary:

Proposition 2.2. For any real subspace arrangement, the sumof Betti numbers of its complement is equal to the sum of Bettinumbers of the complement of its complexification.

As an example, the complement of an arbitrary real hyper-

plane arrangement is just the union of the pieces into which

the hyperplanes cut the Euclidean space, each of these is con-

tractible. Proposition 2.2 implies that the sum of the Betti

numbers of the complement of the complexification of a real

hyperplane arrangement is equal to the number of these pieces.

For instance, as explained above , the sum of the Betti num-

bers of the complement of the real braid arrangement is equal

to d!, and hence, Proposition 2.2 implies that the sum of the

Betti numbers of the complement of the complex braid ar-

rangement is equal to d!. The latter is an interesting and im-

portant topological space studied in particular by Arnold,

[Ar69].

Of course the relation of arrangement theory to combina-

torics is by no means exhausted by the Goresky-MacPherson

formula. For example, a famous result of Orlik and Solomon

connects the cohomology algebra of the complement of a hy-

perplane arrangement to the theory of matroids, see [OS80].

Also the study by DeConcini and Procesi, see [DP95], of res-

olutions of singularities for a collection of divisors, whose in-

tersections are locally arrangements, leads to subtle and inter-

esting combinatorics of nested sets, see [Fei05, FK04, FK05].

One other notable situation is that of toric varieties where

similiar ideas provide connections between algebraic and poly-

hedral geometry, see [Fu93, MS05].

3 Graph homomorphisms and characteristicclasses

Let us now describe one of the many situations in which al-

gebraic topology can be of use for questions in discrete math-

ematics.

A central notion in combinatorics is that of a graph. For

two graphs T and G, a graph homomorphism from T to G is

a set map between vertex sets ϕ : V (T ) →V (G), such that if

x,y ∈V (T ) are connected by an edge, then ϕ(x) and ϕ(y) are

also connected by an edge. An important special case is pro-

vided by considering a graph homomorphism to a completegraph ϕ : G → Kn, which is the same as a vertex coloring of

G with n colors. In particular, the chromatic number of G, de-

noted χ(G), is the minimal n, such that there exists a graph

homomorphism ϕ : G → Kn. It is an important, but difficult

task to compute or to estimate chromatic numbers of graphs.

We mention the notorious 4-coloring theorem as an exam-

ple.

The topological approach to graph colorings was pion-

eered by László Lovász (who is the current IMU President

and has turned 60 this year). In 1978 he used homotopy the-

ory to tackle the problem of determining chromatic numbers

of Kneser graphs, see [Lov78] for details, and also [dL03] for

a nice historical account. Recall that the Kneser graphs are

the graphs defined as follows: for fixed k and n, vertices of

the corresponding Kneser graph are indexed by all k-element

subsets of [n], and two vertices are connected by an edge if

the indexing subsets are disjoint.

We note that if ϕ : T → G and ψ : G → H are graph ho-

momorphisms, then the composition ψ ◦ϕ : T → H is again a

graph homomorphism, and, since identity maps are also graph

homomorphisms, it follows that graphs together with graph

homomorphisms form a category called Graphs.More recently it was realized that the set of all graph ho-

momorphisms between graphs T and G can be enriched to

form a topological space Hom(T,G), see [BK06] for details.

The construction can be rephrased as follows: let ∆V (G) be

a simplex whose set of vertices is V (G), and let C(T,G) de-

note the direct product ∏x∈V (T ) ∆V (G), i.e., the copies of ∆V (G)

are indexed by vertices of T .

Definition 3.1. The complex Hom(T,G) is the subcomplex ofC(T,G) defined by the following condition: σ = ∏x∈V (T) σx ∈Hom(T,G) if and only if for any x,y ∈ V (T ), if (x,y) ∈ E(T ),then (σx,σy) is a complete bipartite subgraph of G.

Here a complete bipartite subgraph of a graph G is a pair

(A,B) of subsets (not necessarily disjoint) of the set of ver-

tices of G, such that any vertex from A is connected by an edge

to any vertex from B. An example of the cell complex of all 3-

colorings of a 6-cycle is given in Figure 1. The reader should

note that several quadrangles have been omitted for the sake

of transparency. There should be three maximal quadrangles

attached to the grey vertex (only the two shaded ones are

shown); the same is the case at the other five vertices at which

the maximal cubes touch each other. In total, there are 18

quadrangles which are not part of higher-dimensional cells.

The topology of Hom(T,G) is inherited from the product

topology of C(T,G), in particular, the cells of Hom(T,G) are

products of simplices. It turned out that when T is an odd

cycle (a polygon) C2r+1, the topology of this space is related

to the chromatic number of G. This is the content of the next

theorem, which was proved in [BK04].

Theorem 3.2. (Lovász Conjecture).

For an arbitrary graph G, and any integers r and k, such thatr ≥ 1, k ≥−1, we have the following implication:if the complex Hom(C2r+1,G) is k-connected, thenχ(G) ≥ k + 4.

Perhaps a useful way of thinking about this and similar re-

sults is to view it as a test: the (higher) connectivity of a cer-

tain topological space is being tested, and if that quantity is

possible to compute (which of course depends heavily on the

space) it provides some information on the chromatic number.

Whenever a loopfree graph T possesses an involution

which flips an edge, the topological space Hom(T,G) has a

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3 12

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Hom(C6,K3)Figure 1. (Essential parts of) the complex Hom (C6,K3)

free involution. This allows us to talk about the Stiefel–Whit-

ney characteristic classes in the cohomology of that space, see

[Ko05a]. It is an intriguing fact that the chromatic number of

G is in fact related to the vanishing of the powers of this char-

acteristic class, and the Lovász Conjecture is one of the in-

dications of that. These connections can be analysed directly,

or by using spectral sequences, we refer to [Ko07, Ko05b] for

further details.

4 A topological approach to evasiveness

In the last section we would like to illustrate ties which ex-

ist between Combinatorial Algebraic Topology and theoret-

ical computer science, more specifically – complexity the-

ory.

Let us fix n and consider the set of all graphs with n ver-

tices. A particular graph is then characterized by the set E(G)considered as a subset of the set of ordered pairs {(i, j) | 1 ≤i < j ≤ n}. Recall that a graph property is called monotone if

the set of graphs which satisfy this property is closed under

removal of edges; examples include planarity and the prop-

erty of being disconnected. Given such a monotone property

G , we can construct a simplicial complex ∆(G ) by taking or-

dered pairs (i, j), for 1 ≤ i < j ≤ n, as vertices, and saying

that a set of such pairs forms a simplex in ∆(G ) if the corre-

sponding graph has the property G .

Let us now consider the following algorithmic situation.

We are given a certain graph property G , which we know,

and a certain graph G, which we do not know. However, we

can ask the oracle questions of the type: Is the edge (i, j) inG? Our task is to decide whether the graph G satisfies the

property G or not. We would like to ask as few questions as

possible. More precisely, we are interested in knowing the so-

called worst-case complexity, that is: how many questions do

we have to ask in the worst case scenario? For example, if the

graph property is trivial (meaning that either no graphs or all

graphs satisfy it), then we do not need to ask any questions

at all. On the other hand, if the property is being a complete

graph, then we might be forced into asking(n

2

)questions: this

would happen if the answers to the first(n

2

)−1 questions are

all positive.

Definition 4.1. A graph property G of graphs on n verticesis called evasive, if we need to ask

(n2

)questions in the worst

case in the course of the algorithm described above; other-wise, we say that G is nonevasive.

Certainly not all non-trivial properties are evasive. Consider

for example the property of being a so-called scorpion graph.

Here a graph G on n vertices is called a scorpion graph if it

has 3 vertices s, t, and b, such that

(1) the vertex s, called the sting, is only connected to vertex t,(2) the vertex t, called the tail, is only connected to vertices s

and b,

(3) the vertex b, called the body, is connected to all vertices,

but s.

This notion is illustrated on Figure 2. It is a nice exercise

in complexity theory to verify that this property is actually

nonevasive.


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s

t

b

Figure 2. An example of a scorpion graph

Clearly, if an edge is removed from a scorpion graph, the

obtained graph does not have to be a scorpion graph anymore:

being a scorpion graph is not a monotone graph property. In

fact, the following is perhaps the most important open ques-

tion pertaining to evasiveness of graphs.

Conjecture 4.2. (Evasiveness Conjecture, also known as Karp

Conjecture)

Every nontrivial monotone graph property for graphs on nvertices is evasive.

It is mind-boggling that so far, the Evasiveness Conjecture

has been verified in the case when n is a prime power, and,

additionally, when n = 6, using topological techniques only;

see [KSS84]. The proof in these cases relies on the interplay

of algebraic invariants of the complex ∆(G ) mentioned above

and the action of the symmetry group Sn and its subgroups.

It is difficult to imagine that whether n is a prime power or

not is essential for the validity of the Evasiveness Conjecture,

and it is still not understood what topology has to do with it.

AcknowledgmentsThe author would like to thank Martin Raussen for many

comments which helped to improve the presentation substan-

tially.

Bibliography

[Ar69] V.I. Arnold, The cohomology ring of the group of dyedbraids, Mat. Zametki 5, (1969), 227–231. (Russian)

[BK04] E. Babson, D.N. Kozlov, Proof of the Lovász Conjecture,

Annals of Mathematics (2) 165 (2007), no. 3, 965–1007.

[BK06] E. Babson, D.N. Kozlov, Complexes of graph homomor-phisms, Israel J. Math., 152 (2006), 285–312.

[DP95] C. De Concini, C. Procesi, Wonderful models of sub-

space arrangements, Selecta Math. (N.S.) 1 (1995), no.

3, 459–494.

[Fei05] E.-M. Feichtner, De Concini-Procesi wonderful arrange-ment models: a discrete geometer’s point of view, in:

Combinatorial and computational geometry, Math. Sci.

Res. Inst. Publ. 52, Cambridge Univ. Press, Cambridge,

2005, pp. 333–360.

[FK04] E.-M. Feichtner, D.N. Kozlov, Incidence combinatoricsof resolutions, Selecta Math. (N.S.) 10 (2004), no. 1, pp.

37–60.

[FK05] E.-M. Feichtner, D.N. Kozlov, A desingularization ofreal differentiable actions of finite groups, Int. Math. Res.

Not. 2005, no. 15, pp. 881–898.

[Fu93] W. Fulton, Introduction to toric varieties, Annals of

Mathematics Studies 131, The William H. Roever Lec-

tures in Geometry, Princeton University Press, Princeton,

NJ, 1993, xii+157 pp.

[GoM88] M. Goresky, R. MacPherson, Stratified Morse Theory,Ergebnisse der Mathematik und ihrer Grenzgebiete, vol.

14, Springer-Verlag, Berlin Heidelberg New York, 1988.

[KSS84] J. Kahn, M. Saks, D. Sturtevant, A topological approachto evasiveness, Combinatorica 4 (1984), 297–306.

[Ko05a] D.N. Kozlov, Chromatic numbers, morphism complexes,and Stiefel–Whitney characteristic classes, in: GeometricCombinatorics (eds. E. Miller, V. Reiner, B. Sturmfels),

IAS/Park City Mathematics Series 14, American Math-

ematical Society, Providence, RI; Institute for Advanced

Study (IAS), Princeton, NJ, 2007; pp. 249–316.

[Ko05b] D.N. Kozlov, Cohomology of colorings of cycles, Amer-

ican J. Math., in press. arXiv:math.AT/0507117[Ko07] D.N. Kozlov, Combinatorial Algebraic Topology, Algo-

rithms and Computation in Mathematics 21, Springer-

Verlag Berlin Heidelberg, 2008, xx+390 pp., 115 illus.

[dL03] M. de Longueville, 25 Jahre Beweis der Kneservermu-tung. Der Beginn der topologischen Kombinatorik, (Ger-

man) [25th anniversary of the proof of the Kneser con-jecture. The start of topological combinatorics], Mitt.

Deutsch. Math.-Ver. 2003, no. 4, pp. 8–11.

[Lov78] L. Lovász, Kneser’s conjecture, chromatic number, andhomotopy, J. Combin. Theory Ser. A 25, (1978), no. 3,

319–324.

[McC01] J. McCleary, A user’s guide to spectral sequences, Sec-

ond edition, Cambridge Studies in advanced mathemat-

ics 58, Cambridge University Press, Cambridge, 2001.

[MS05] E. Miller, B. Sturmfels, Combinatorial Commutative Al-gebra, Graduate Texts in Mathematics 227, Springer-

Verlag, New York, 2005, xiv+417 pp.

[OS80] P. Orlik, L. Solomon, Combinatorics and Topology ofcomplements of hyperplanes, Invent. Math. 56 (1980),

167–189.

Dmitry N. Kozlov [[email protected]]The author is recipient of the WallenbergPrize of the Swedish Mathematics Society(2003), the Gustafsson Prize of the GöranGustafsson Foundation (2004), and the Euro-pean Prize in Combinatorics (2005). He hasalso authored a book on the topic which has

recently appeared in Springer-Verlag. His research lies at theinterface of Discrete Mathematics, Algebraic Topology, andTheoretical Computer Science.He has obtained his doctorate from the Royal Institute ofTechnology, Stockholm in 1996. After longer stays at theMathematical Sciences Research Institute at Berkeley, theMassachusetts Institute of Technology, the Institute for Ad-vanced Study at Princeton, the University of Washington,Seattle, and Bern University, he has been a Senior Lecturerat the Royal Institute of Technology, Stockholm, and an Assis-tant Professor at ETH Zurich. Currently he holds the Chair ofAlgebra and Geometry at the University of Bremen, Germany.

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On PlatonismReuben Hersh (University of New Mexico, USA)

“Platonism” can mean a lot of differ-ent things. Even “Platonism in math-ematics” can mean a lot of different things. In my writings on the philoso-phy of mathematics, I have been con-cerned about the philosophical stance or preconceptions of practicing math-ematicians, whether explicitly formu-lated or not. As I have written, this

usually involves some choice or combination or alter-nation of “formalism” and “Platonism”, both of them in rough-and-ready, naïve versions. Their “Platonism” says mathematical objects exist independently of our knowl-edge or activity, and mathematical truth is objective, with the same status as scientifi c truth about the physi-cal world. This may be boiled down to the phrase “out there.” That’s where mathematical entities are, meaning, not “in here.”

As for “formalism”, that is taken to mean that math is no more than logical deduction from formally stated “ax-ioms,” so that the “facts” about mathematical objects are nothing more than statements that one formal sentence follows logically from some other formal sentence; there is no “meaning” or “reference,” just logical manipulation of logical formulas.

These two points of view are incompatible, yet it seems that most mathematicians oscillate between the two. My purpose in writing about this has been to argue that both notions are false. Formalism is wildly incompat-ible with the actual practice of mathematics, whether in research or in teaching. Platonism is incompatible with the standard view of the nature of reality, as held by the majority of scientists and mathematicians: there is only one universe, one real world, which is physical reality, in-cluding its elaboration into the realm of living things, and elaborated from there into the realm of humankind with its social, cultural, and psychological aspects. I have ar-gued that these social, cultural and psychological aspects of humankind are real, not illusory or negligible, and that the nature of mathematical truth and existence is to be understood in that realm, rather than in some other inde-pendent “abstract” reality.

My view of Platonism – always referring to the com-mon, every-day Platonism of the typical working math-ematician – is that it expresses a correct recognition that there are mathematical facts and entities, that these are not subject to the will or whim of the individual mathe-matician but are forced on him as objective facts and en-tities which he must learn about and whose independent existence and qualities he seeks to recognize and discov-er. The fallacy of Platonism is in the misinterpretation of this objective reality, putting it outside of human culture and consciousness. Like many other cultural realities, it is external, objective,

from the viewpoint of any individual, but internal, histori-cal, socially conditioned, from the viewpoint of the society or the culture as a whole.

(One might note, in passing, that other socio-cultural phenomena, like the divine right of kings, the innate natu-ral inferiority of slaves, etc etc have also been regarded for centuries as part of the objective order of the Universe.)

The recent article by Davies1 declares death to Pla-tonism, on the grounds of neurophysiological discoveries about arithmetical and geometrical thinking. Researchers are now able to observe blood fl ow into specifi c sections of the brain associated with various mental processes, in-cluding mathematical ones. These discoveries add a lot of empirical weight to our previous general belief, that think-ing is something that happens mostly in the brain. What new philosophical insights do we gain? Well, Platonism requires some mental faculty by which the mathematician can connect to the abstract external non-physical realm where the objects he is studying are supposed to exist. This faculty can be labeled “intuition,” but such a label is not an explanation. In fact, the dualism that is forced by Platonism – the division of reality into two realms, one physical, including the brain of the mathematician, and the other “abstract,” “out there,” neither physical nor so-cial nor psychological, just “abstract” and “out there” – is a fatal fl aw, much the same as the fl aw in Descartes’ dual-ism of mind and matter. It seems that Davies regards the evidence that thinking takes place in the brain as proof that there is no such “intuition”, in the sense of a spe-cial mental faculty for connecting to “out there.” But with or without neurophysiological evidence, it is pretty clear that the posited “intuition” is an ad hoc artifact, lack-ing any specifi city or clear description, let alone empiri-cal evidence. It is nothing more than something that, for the Platonist, “has to” be available to the mathematician, since he/she evidently is able to discover mathematical facts, and these facts are “out there,” not “in here.”

If by Platonism one simply means the rejection of for-malism – the conviction that the things we are studying are real, and that they have objective properties which we can discover – then there is no doubt that this con-viction is supportive, indeed perhaps even necessary, for mathematical research. While one is proving a theorem or fi nding a counterexample, one is not in the mode of considering the statement of the theorem or counterex-ample meaningless! But it is not necessary, either logi-cally or psychologically, to locate this meaning in some eternal, inhuman, abstract reality.

I once took a vote in a talk at New Mexico State Uni-versity in Las Cruces. The question was, “Was the spectral theorem on self-adjoint operators in Hilbert space true before the Big Bang, before there was a universe?” The vote was yes, by a margin of 75 to 25. But there were no self-adjoint operators, no Hilbert space, before the twenti-eth century! So no statement about them could be true or false or meaningful (it could not refer to anything) before these concepts had been established in the consciousness of mathematicians. However, it is true that mathematical

1 Let Platonism die, EMS-Newsletter 64 (2007), 24–25.


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theories like the theory of self-adjoint operators in Hilbert space, while they come into existence historically (and can even go out of existence historically) are time-independ-ent, in the sense that they do not refer to or in their con-tent include or make note of any non-mathematical, exter-nal data, including the particular century of their birth. We construct mathematics without reference to non-math-ematical data, including spatial (in what country was the mathematician who proved the theorem?) and temporal. The spectral theorem’s content or reference excludes any spatial-temporal data of ordinary daily life.

I have compared this to a movie, which we know is just shadows and colors projected on a screen. This fac-tual knowledge in no way confl icts with seeing the movie as it is intended to be seen, as a (fi ctional) reality which we fully understand, participate in, and can reason about. There are objective facts relating to the content of the movie, without relegating it to an abstract realm “out there.” At the same time, the movie has a physical reality in terms of lights and shadows, and cannot exist except on the basis of this or some other comparable physical real-ity. Similarly, the content of mathematical theorems and concepts is real, we do have reliable knowledge of it. That content is part of human culture and consciousness, and is contingent on the existence and activity of the species homo sapiens. With this understanding, full acceptance and acknowledgment of the reality and meaningfulness of mathematics does not contradict or confl ict with the ordinary scientifi c view of reality. The mental, social and cultural, including the mathematical, are grounded in the physical – the fl esh and blood of past and present hu-mans, especially mathematicians. We can recognize this, even while the detailed nature of this grounding – just how our thoughts are carried out by our brains – may never be completely understood.

The puzzle for us to resolve is the universality of mathematics. Unlike other cultural realms, there is a universality or commonality of mathematical facts – the “laws” of arithmetic, for instance – which is strikingly dif-ferent from the great variations in other cultural realms – language, religion, music, family structure, etc, in differ-ent cultures around the world. Why is this? Well, to begin with, the facts of elementary arithmetic, the small natural numbers, are physical facts, they describe the behavior of collections of stable, non-interacting, discrete objects like coins or buttons. These are as independent of culture as other parts of physics, like the fact that a stone will fall to the ground. The ability to develop logical consequences also seems to be to a great extent culture – independent, which suggests a possible basis in our brain structures, just as linguists have postulated some basis in our brain struc-tures for the universal features that all human languages seem to share. Certainly, along with physical reality, we must consider the special features of the human brain, if we want to understand more deeply the nature of math-ematics. This is a major project for empirical science – not for philosophy! From this point of view, the facts present-ed by Davies are relevant and interesting. They do not refute Platonism, they are part of the scientifi c program that one focuses on after rejecting Platonism.

To put the matter as simply as possible: everyone agrees that mathematics is a historically derived cultural activity. I say, there is no need to give it any other “meta-physical” description. As a cultural activity it has certain puzzling features: its seeming “certainty,” and “universal-ity.” Platonists account for this puzzle by postulating a special existential realm, with matching mental faculty (“intuition”). They simply disregard the strange incom-patibility of such a realm and faculty with our common understandings of both philosophy of science and phi-losophy of mind. I consider it more promising to regard the nature of mathematics as a problem for a multi-dis-ciplinary empirical investigation, analogous, for example, to the nature of language, where an empirical science, linguistics, is thriving. The best example of such investi-gation is the fascinating work of Stanislas Dehaene [D]. My recent edited book was intended as a contribution to this enterprise [H1].

My most persistent opponent has been Martin Gard-ner, who is a theist, a believer in the effi cacy of prayer, and who has declared that I am on the “slippery slope” to solipsism (in one review) and to Stalinism (in another review). Others have seen a resemblance to Jung’s “col-lective unconscious”. I see nothing Jungian here, but Jungianism is not so bad, compared to solipsism and Stalinism! I have also been honored by a critique from William Tait, who is described on the jacket of his book as “one of the most eminent philosophers of mathemat-ics of his generation” [T]. He fi nds my description of Platonism rather ridiculous, and seems not to know the difference between my saying what some other people seem to think and saying what I myself think. (He pro-poses a distinction between “realism” and what he calls “hyperrealism.” “Realism” is OK, says William Tait. The absurdities of Platonism, he explains, are attributes of “hyperrealism.”)

References[D] Dehaene, S. The Number Sense, Oxford University Press 1997[H1] Hersh, R. What is Mathematics, Really? Oxford University Press

1997[H2] Hersh, R. (Ed.) 18 Unconventional essays on the nature of math-

ematics, Springer, 2006[T] Tait, W. The Provenance of Pure Reason, Oxford University Press,

2005

Reuben Hersh [[email protected]] has published research on partial dif-ferential equations, random evolutions, and probability. He lives in Santa Fe, New Mexico, in the USA. He was a stu-dent of Peter Lax, and a co-author with

Phil Davis, Martin Davis, Paul Cohen, Richard J. Griego, Jim Donaldson, Tosio Kato, George Papanicolaou, Mark Pinsky, Priscilla Greenwood, and others. With Vera John-Steiner, he has a forthcoming book entitled “Loving and Hating Mathematics: Inside Mathematical Life.”

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Mathematical Platonism and its OppositesBarry Mazur (Harvard University, Cambridge MA, USA)

We had the sky up there, all speckled with stars, and we used to lay on our backs and look up at them, and discuss about whether they was made or only just happened – Jim he allowed they was made, but I allowed they happened; I judged it would have took too long to make so many.

mused Huckleberry Finn. The analogous query that mathematicians continually fi nd themselves confronted with when discussing their art with people who are not mathematicians is:

Is mathematics discovered or invented?I will refer to this as The Question, acknowledging

that this fi ve-word sentence, ending in a question mark – and phrased in far less contemplative language than that used by Huck and Jim – may open conversations, but is hardly more than a token, standing for puzzlement regarding the status of mathematics.

If you engage in mathematics long enough, you bump into The Question, and it won’t just go away.1 If we wish to pay homage to the passionate felt experience that makes it so wonderful to think mathematics, we had bet-ter pay attention to it.

Some intellectual disciplines are marked, even scarred, by analogous concerns. Anthropology, for example has a vast, and dolefully introspective, literature dealing with the conundrum of whether we can ever avoid – wittingly or unwittingly – clamping the templates of our own cul-ture onto whatever it is we think we are studying: how much are we discovering, how much inventing?

Such a discovered/invented perplexity may or may not be a burning issue for other intellectual pursuits, but it burns exceedingly bright for mathematics, and with a strangeness that isn’t quite matched when it pops up in other fi elds. For example, if you were to say “Priestley discovered oxygen but Lavoisier invented it” I think I know roughly what you mean by that utterance, without our having to synchronize our private vocabularies ter-ribly much. But to intelligently comprehend each other’s possibly differing attitudes towards circles, triangles, and numbers, we would also have to come to some – albeit ever-so-sketchy – understanding of how we each view, and talk about, a lot more than mathematics.2

For me, at least, the anchor of any conversation about these matters is the experience of doing mathe-matics, and of groping for mathematical ideas. When I read literature that is ostensibly about The Question, I ask myself whether or not it connects in any way with my felt experience, and even better, whether it reveals something about it. I’m often – perhaps always – disap-

pointed. The bizarre aspect of the mathematical experi-ence – and this is what gives such fi erce energy to The Question – is that one feels (I feel) that mathematical ideas can be hunted down, and in a way that is essen-tially different from, say, the way I am currently hunt-ing the next word to write to fi nish this sentence. One can be a hunter and gatherer of mathematical concepts, but one has no ready words for the location of the hunt-ing grounds. Of course we humans are beset with illu-sions, and the feeling just described could be yet another. There may be no location.

There are at least two standard ways of – if not ex-actly answering, at least – fi elding The Question by offer-ing a vocabulary of location. The colloquial tags for these locations are In Here and Out There (which seems to me to cover the fi eld).

The fi rst of these standard attitudes, the one with the logo In Here – which is sometimes called the Kantian (poor Kant!) – would place the source of mathematics squarely within our faculties of understanding. Of course faculties (Vermögen) and understanding (Verstand) are loaded eighteenth century words and it would be good – in this discussion at least – to disburden ourselves of their baggage as much as possible. But if this camp had to choose between discovery and invention, those two too-brittle words, it would opt for invention.

The “Out There” stance regarding the discovery/in-vention question whose heraldic symbol is Plato (poor Plato!) is to make the claim, starkly, that mathematics is the account we give of the timeless architecture of the cosmos. The essential mission, then, of mathematics is the accurate description, and exfoliation, of this architecture. This approach to the question would surely pick discov-ery over invention.

Strange things tend to happen when you think hard about either of these preferences.

For example, if we adopt what I labelled the Kantian position we should keep an eye on the stealth word “our” in the description of it that I gave, hidden as it is among behemoths of vocabulary (Vermögen, Verstand). Exactly whose faculties are being described? Who is the we? Is the we meant to be each and every one of us, given our separate and perhaps differing and often faulty faculties? If you feel this to be the case, then you are committed to viewing the mathematical enterprise to be as variable as humankind. Or are you envisioning some sort of distil-late of all actual faculties, a more transcendental faculty, possessed by a kind of universal or ideal we, in which

1 Garrison Keillor, a wonderful radio raconteur has in his rep-ertoire a fi ctional character, Guy Noir, who tangles indefati-gably with “life’s persistent questions.” This is all to the good. We should pay particular honour to the category of persist-ent questions even though – or, especially because – those are the chestnuts that we’ll never crack.

2 For a start: you and I turn adjectives into nouns (red cows → red; fi ve cows → fi ve) with only the barest fl ick of a thought. What is that fl ick? Understanding the differences in our sense of what is happening here may tell us lots about our differences regarding matters that can only be discussed with much more mathematical vocabulary.


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case the Kantian view would seem to merge with the Pla-tonic.3

If we adopt the Platonic view that mathematics is dis-covered, we are suddenly in surprising territory, for this is a full-fl edged theistic position. Not that it necessarily posits a god, but rather that its stance is such that the only way one can adequately express one’s faith in it, the only way one can hope to persuade others of its truth, is by abandoning the arsenal of rationality, and relying on the resources of the prophets.

Of course, professional philosophers are in the busi-ness of formulating anti-metaphysical or metaphysical positions, decorticating them, defending them, and refut-ing them.4 Mathematicians, though, may have another – or at least a prior – duty in dealing with The Question. That is, to be meticulous participant/observers, faithful to the one aspect of The Question to which they have sole proprietary rights: their own imaginative experience. What, precisely, describes our inner experience when we (and here the we is you and me) grope for mathematical ideas? We should ask this question open-eyed, allowing for the possibility that whatever it is we experience may delude us into fabricating ideas about some larger frame-work, ideas that have no basis.5

I suspect that many mathematicians are as unsatisfi ed by much of the existent literature about The Question as I am. To be helpful here, I’ve compiled a list of Do’s and Don’t’s for future writers promoting the Platonic or the Anti-Platonic persuasions.

For the PlatonistsOne crucial consequence of the Platonic position is that it views mathematics as a project akin to physics, Pla-tonic mathematicians being – as physicists certainly are – describers or possibly predictors – not, of course, of the physical world, but of some other more noetic6 en-tity. Mathematics – from the Platonic perspective – aims, among other things, to come up with the most faithful description of that entity.

This attitude has the curious effect of reducing some of the urgency of that staple of mathematical life: rigor-ous proof. Some mathematicians think of mathematical proof as the certifi cate guaranteeing trustworthiness of, and formulating the nature of, the building-blocks of the edifi ces that comprise our constructions. Without proof: no building-blocks, no edifi ce. Our step-by-step articulat-ed arguments are the devices that some mathematicians feel are responsible for bringing into being the theories we work in. This can’t quite be so for the ardent Platon-ist, or at least it can’t be so in the same way that it might be for the non-Platonist. Mathematicians often wonder about – sometimes lament – the laxity of proof in the physics literature. But I believe this kind of lamentation is based on a misconception, namely the misunderstand-ing of the fundamental function of proof in physics. Proof has principally (as it should have, in physics) a rhetori-cal role: to convince others that your description holds together, that your model is a faithful re-production, and possibly to persuade yourself of that as well. It seems to me that, in the hands of a mathematician who is a de-

termined Platonist, proof could very well serve primarily this kind of rhetorical function – making sure that the de-scription is on track – and not (or at least: not necessar-ily) have the rigorous theory-building function it is often conceived as fulfi lling.

My feeling, when I read a Platonist’s account of his or her view of mathematics, is that unless such issues re-garding the nature of proof are addressed and conscien-tiously examined, I am getting a superfi cial account of the philosophical position, and I lose interest in what I am reading.

But the main task of the Platonist who wishes to per-suade non-believers is to learn the trade, from prophets and lyrical poets, of how to communicate an experience that transcends the language available to describe it. If all you are going to do is to chant credos synonymous with “the mathematical forms are out there,” – which some proud essays about mathematical Platonism con-tent themselves to do – well, that will not persuade.

For the Anti-PlatonistsHere there are many pitfalls. A common claim, which is meant to undermine Platonic leanings, is to introduce into the discussion the theme of mathematics as a human, and culturally dependent pursuit and to think that one is actually conversing about the topic at hand. Consider this, though: If the pursuit were writing a description of the Grand Canyon and if a Navajo, an Irishman, and a Zoroastrian were each to set about writing their descrip-tions, you can bet that these descriptions will be cultur-ally-dependent, and even dependent upon the moods and education and the language of the three describers. But my having just recited all this relativism regarding the three descriptions does not undermine our fi rm faith in the existence of the Grand Canyon, their common fo-cus. Similarly, one can be the most ethno-mathematically

3 A more general lurking question is exactly how we are to view the various ghosts in the machine of Kantian idealism – for example, who exactly is that little-described player haunt-ing the elegant concept of universally subjective judgments and going under a variety of aliases: the sensus communis or the allgemeine Stimme?

4 A very useful – and to my mind, fi ne – text that does exactly this type of lepidoptery is Mark Balaguer’s Platonism and Anti-Platonism in Mathematics, Oxford Univ. Press (1998).

5 When I’m working I sometimes have the sense – possibly the illusion – of gazing on the bare platonic beauty of structure or of mathematical objects, and at other times I’m a happy Kantian, marvelling at the generative power of the intui-tions for setting what an Aristotelian might call the formal conditions of an object. And sometimes I seem to straddle these camps (and this represents no contradiction to me). I feel that the intensity of this experience, the vertiginous im-aginings, the leaps of intuition, the breathlessness that results from “seeing” but where the sights are of entities abiding in some realm of ideas, and the passion of it all, is what makes mathematics so supremely important for me. Of course, the realm might be illusion. But the experience?

6 “inner knowing”, a kind of intuitive consciousness – direct and immediate access to knowledge beyond what is available to our normal senses and the power of reason.

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conscious mathematician on the globe, claiming that all our mathematical scribing is as contingent on ephemeral circumstance as this morning’s rain, and still one can be the most devout of mathematical Platonists.

Now this pitfall that I have just described is harm-less. If I ever encounter this type of mathematics-is-a-hu-man-activity argument when I read an essay purporting to defuse, or dispirit, mathematical Platonism I think to myself: human activity! what else could it be? I take this part of the essay as being irrelevant to The Question.

A second theme that seems to have captured the im-agination of some anti-Platonists is recent neurophysio-logical work – a study of blood fl ow into specifi c sections of the brain – as if this gives an insider’s view of things.7 Well, who knows? Neuro-anatomy and chemistry have been helpful in some discussions, and useless in others . To show this theme to be relevant would require a precisely argued explanation of exactly how blood fl ow patterns can refute, or substantiate, a Platonist – or any – disposi-tion. A satisfying argument of that sort would be quite a marvel! But just slapping the words blood fl ow – as if it were a poker-hand – onto a page doesn’t really work.

Sometimes the mathematical anti-Platonist believes that headway is made by showing Platonism to be unsup-portable by rational means, and that it is an incoherent position to take when formulated in a propositional vo-cabulary.

It is easy enough to throw together propositional sen-tences. But it is a good deal more diffi cult to capture a Pla-tonic disposition in a propositional formulation that is a full and honest expression of some fl esh-and-blood math-ematician’s view of things. There is, of course, no harm in

7 like the old Woody Allen movie Everything you wanted to know about sex but were afraid to ask

trying – and maybe a good exercise. But even if we clev-erly came up with a proposition that is up to the task of ex-pressing Platonism formally, the mere fact that the propo-sition cannot be demonstrated to be true won’t necessarily make it vanish. There are many things – some true, some false – unsupportable by rational means. For example, if you challenge me to support – by rational means – my claim that I dreamt of Waikiki last night, I couldn’t.

So, when is there harm? It is when the essayist becomes a leveller. Often this happens when the author writes ex-tremely well, super coherently, slowly withering away the Platonist position by – well – the brilliant subterfuge of making the whole discussion boring, until I, the reader, become convinced – albeit momentarily, within the frame-work of my reading the essay – that there is no “big deal” here: the mathematical enterprise is precisely like any oth-er cultural construct, and there is a fallacy lurking in any claim that it is otherwise. The Question is a non-question.

But someone who is not in love won’t manage to de-fi nitively convince someone in love of the non-existence of eros; so this mood never overtakes me for long. Hap-pily I soon snap out of it, and remember again the re-markable sense of independence – autonomy even – of mathematical concepts, and the transcendental quality, the uniqueness – and the passion – of doing mathematics. I resolve then that (Plato or Anti-Plato) whatever I come to believe about The Question, my belief must thorough-ly respect and not ignore all this.

Barry Mazur [[email protected]] is the Gerhard Gade University Professor at Harvard University.

EMS Tracts in Mathematics Vol. 5

Gennadiy Feldman (National Academy of Sciences, Kharkov, Ukraine)Functional Equations and Characterization Problems on Locally Compact Abelian Groups

ISBN 978-3-03719-045-6 April 2008, 268 pages, hardcover, 17 x 24 cm. 58.00 Euro

This book deals with the characterization of probability distributions. It is well known that both the sum and the difference of two Gaussian independent random variables with equal variance are independent as well. The converse statement was proved independently by M. Kac and S. N. Bernstein. This result is a famous example of a characterization theorem. In general, characterization problems in mathematical statistics are statements in which the description of possible distributions of random variables follows from properties of some functions in these variables.

In recent years, a great deal of attention has been focused upon generalizing the classical characterization theorems to random variables with values in various algebraic structures such as locally compact Abelian groups, Lie groups, quantum groups, or symmetric spaces. The present book is aimed at the generalization of some well-known characterization theorems to the case of independent random variables taking values in a locally compact Abelian group X. The main attention is paid to the characterization of the Gaussian and the idempotent distribution (group analogs of the Kac–Bernstein, Skitovich–Darmois, and Heyde theorems). The solution of the corresponding problems is reduced to the solution of some functional equations in the class of continuous positive defi nite functions defi ned on the character group of X. Group analogs of the Cramér and Marcinkiewicz theorems are also studied.

The author is an expert in algebraic probability theory. His comprehensive and self-contained monograph is addressed to mathematicians working in probability theory on algebraic structures, abstract harmonic analysis, and functional equations. The book concludes with comments and unsolved problems that provide further stimulation for future research in the theory.

European Mathematical Society Publishing HouseSeminar for Applied Mathematics, ETH-Zentrum FLI C4

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Encyclopedia of OptimizationC. A. Floudas, Princeton University, NJ, USA; P. M. Pardalos, University of Florida, Gainesville, USA (Eds.)

From the reviews of the First Edition The five volumes, plus an Index, of the Encyclopedia of Optimization are a major resource for anyone interested in optimization and its many applica-tions . . . these volumes will enable the user to quickly understand and make use of specific methods of optimization, and to apply them to applications of interest.... Professor J.B. Rosen,Computer Science Department, University of California at San Diego

The goal of the Encyclopedia of Optimization is to introduce the reader to a complete set of topics that show the spectrum of research, the richness of ideas, and the breadth of applications that has come from this field.In 2000, the first edition was widely acclaimed and received high praise. J.B. Rosen crowned it “an indepensible resource” and Dingzhu Du lauded it as “the standard most important reference in this very dynamic research field.” Top authors such as Herbert Hauptman (winner of the Nobel Prize) and Leonid Khachiyan (the Ellipsoid theorist) contributed and the second edition keeps these seminal entries.The second edition builds on the success of the former edition with more than 150 completely new entries, designed to ensure that the reference addresses recent areas where optimization theories and techniques have advanced. Particularly heavy attention resulted in health science and transporta-tion, with entries such as “Algorithms for Genomics,” “Optimization and Radiotherapy Treatment Design,” and “Crew Scheduling.”

Print2nd ed. 2008. Approx. 4000 p. Hardcover.ISBN 978-0-387-74758-3 € 1699,00 | £1307.00Prepublication price € 1359,00 | £1045.50

eReferenceISBN 978-0-387-74759-0 € 1699,00 | £1307.00Prepublication price € 1359,00 | £1045.50

Print + eReferenceISBN 978-0-387-74760-6 € 2129,00 | £1637.50Prepublication price € 1699,00 | £1307.00

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Random CurvesJourneys of a Mathematician

N. Koblitz, University of Washington, Seattle, WA, USA

Neal Koblitz is a co-inventor of one of the two most

popular forms of encryption and digital signature, and his autobiographical memoirs are collected in this volume. Besides his own personal career in mathematics and cryptography, Koblitz details his travels to the Soviet Union, Latin America, Vietnam and elsewhere; political activism; and academic controversies relating to math education, the C. P. Snow “two-culture“ problem, and mistreatment of women in academia. These engaging stories fully capture the experiences of a student and later a scientist caught up in the tumultuous events of his generation.

2008. X, 392 p. 28 illus., 18 in color. HardcoverISBN 978-3-540-74077-3 € 34,95 | £24.50

Mathematical EpidemiologyF. Brauer, University of British Columbia, Vancouver, Canada; P. van den Driessche,University of Victoria, Canada; J. Wu, York Univer-sity, Toronto, Canada (Eds.)

Based on lecture notes of two summer schools with a mixed audience from mathematical sciences, epidemiology and public health, this volume offers a comprehensive introduction to basic ideas and techniques in modeling infectious diseases, for the comparison of strategies to plan for an anticipated epidemic or pandemic, and to deal with a disease outbreak in real time.

2008. XVIII, 408 p. 71 illus., 27 in color. (Lecture Notes in Mathematics / Mathematical Biosciences Subseries, Volume 1945) SoftcoverISBN 978-3-540-78910-9 € 69,95 | £54.00

From Hyperbolic Systems to Kinetic TheoryA Personalized Quest

L. Tartar, Carnegie Mellon University, Pittsburgh, PA, USA

2008. XXVIII, 282 p. (Lecture Notes of the Unione Matematica Italiana, Volume 6) SoftcoverISBN 978-3-540-77561-4 € 49,95 | £38.50

Multiscale MethodsAveraging and Homogenization

G. A. Pavliotis, Imperial College London, UK; A. M. Stuart, University of Warwick, Coventry, UK

This introduction to multiscale methods gives you a broad overview of the methods’ many uses and applications. The book begins by setting the theoretical foundations of the methods and then moves on to develop models and prove theorems. Extensive use of examples shows how to apply multiscale methods to solving a variety of problems. Exercises then enable you to build your own skills and put them into practice. Extensions and generalizations of the results presented in the book, as well as references to the literature, are provided in the Discussion and Bibliography section at the end of each chapter.With the exception of Chapter One, all chapters are supplemented with exercises.

2008. XVIII, 310 p. (Texts in Applied Mathematics, Volume 53) HardcoverISBN 978-0-387-73828-4 € 39,95 | £30.50

Piecewise-smooth Dynamical SystemsTheory and Applications

M. d. Bernardo, University of Bristol, UK; C. Budd,University of Bath, UK;

A. Champneys, P. Kowalczyk, University of Bristol, UK

This book presents a coherent framework for understanding the dynamics of piecewise-smooth and hybrid systems. An informal introduction expounds the ubiquity of such models via numerous. The results are presented in an informal style, and illustrated with many examples. The book is aimed at a wide audience of applied mathematicians, engineers and scientists at the beginning postgraduate level. Almost no mathematical background is assumed other than basic calculus and algebra.

2008. XXII, 483 p. 234 illus. (Applied Mathematical Sciences, Volume 163) HardcoverISBN 978-1-84628-039-9 € 76,95 | £49.00

New from Springer

History


The meeting at Wernigerode Arild Stubhaug (Hyggen, Norway)

The following is an excerpt from Arild Stubhaug’s bio-graphy of the Swedish mathematician Mittag-Leffl er (1846–1927) chosen by the author himself and translat-ed by Ulf Persson, who is one of the editors of the EMS Newsletter. The biography (to be reviewed elsewhere in this issue) has appeared in its original Norwegian and as a Swedish translation (by K.O. Widman) last year. An English edition in preparation with Springer-Verlag and due 2009 (independent of the excerpt below).

The excerpt takes place in August 1888 in the Harz Mountains in Northern Germany, still a popular ‘Spa-type’ destination. Wernigerode is a small, touristy town situated in the north, sporting medieval half-timbered houses and a castle with a wide view of the surroundings. The visitor will fi nd here an ample offering of both culture and nature but I doubt any references to its mathematical history.

Since the beginning of July 1888, Weierstrass had been staying in Wernigerode in the Harz. He rented a number of rooms at Müllers Hotel for himself and his two sisters. Sonja Kovalevsky had also come here from London and Paris. She was putting the fi nal touches to the treatise that would eventually get her the Bordin prize and she

hoped that by being around Weierstrass she would be able to conclude her work.

On 3 August Mittag-Leffl er arrived in Wernigerode, or rather at a station one hour away by train from it, where he was met and welcomed by Weierstrass and Sonja. Weierstrass looked well and Sonja ‘smashing’, he noted in his diary. For Mittag-Leffl er, the main purpose of meeting Weierstrass was to discuss and assess the twelve papers submitted for the mathematical prize of Oscar II. But he also entertained plans of asking Weierstrass to present his latest results so that they would not be lost to posterity should Weierstrass not fi nd the time to edit them.

When he told Sonja about his plan the fi rst evening, she implored him to abstain. She badly needed Weierstrass’ time and attention in order to complete her own work. Besides the old master had solemnly asked her to come so he could relay to her what he no longer felt able to edit himself.

The next day Mittag-Leffl er noted in his diary: ‘Math-ematical discussions [with Weierstrass and Sonja] of high interest’. In the evening Volterra also arrived at Werni-gerode and for the next few days, Weierstrass divided his time between his three former students. In the morning he listened to and commented on Volterra’s generaliza-tions of Abel’s theorem for multiple integrals; in the af-ternoon he expounded at length on the three-body prob-lem to the other two. In addition they went on shorter or longer walks, sometimes all together and sometimes two and two. Weierstrass expressed to Mittag-Leffl er his concern of whether Sonja would get her work completed – so much still remained in order to bring it into shape.

When Volterra left Wernigerode after three days, it was cold and raining. Mittag-Leffl er took some short walks, fi rst with Weierstrass and then with Sonja. Sonja told him about her relationship with Maxim Kovalevsky – that she did not want to marry him but would rather live with him clandestinely. In this way she hoped to have a hold on him and make him remain the ‘devoted lover’ she had always wanted to have. Maxim on the other hand had always claimed that anything but marriage would denigrate her. But such a view, she thought, was just brought about by his vanity; the real reason he wanted her as a wife was just to be able to keep mistresses on the side. On top of everything he would rather be the most famous of the two. She continued to talk about wanting to be a mistress, not a wife, and thus it was sad to real-ize that men only respected her in such a way that the mere thought of her as a mistress appeared impossible. And when she started to suggest that in the vacations she would not mind having one, or better still several intel-lectually well-endowed men as her lovers, while during the semesters she would rather prefer to be left alone

Karl Weierstrass (1815–1897) © Institut Mittag-Leffl er

Sonja Kovalevsky (1850–1891) © Institut Mittag-Leffl er

Gösta Mittag-Leffl er (1846–1927) © Institut Mittag-Leffl er

Vito Volterra (1860–1940)

History


and work, Mittag-Leffl er could not help but remark that ‘this is not how it usually comes across during the ac-tual semesters’ and he added that unfortunately for her she lacked those physical attributes a man demands of a mistress. On the other hand she was probably richly equipped with those attributes a man would want in a wife. ‘But unhappy the man who had her for his wife,’ he added because the egotism Sonja had developed to such a high degree and her indifference, concealed under a lively and engaging temperament, would rather soon bring a husband to the end of his wits. ‘With her, every-thing is head and imagination but her heart has not yet vibrated’, he wrote, and thought of her terrible temper and the scenes she was capable of creating, as well as her total absence of any sense of duty and responsibil-ity, which she so openly admitted to. He concluded that Sonja was probably completely in the right. It would be best for her not to get married at all.

About a week after Mittag-Leffl er’s arrival in Werni-gerode, Cantor came as well. In the meantime both pro-fessor Tietgen from Berlin and professor Hettner, who did not live too far away, had dropped by.

Cantor turned out to prove himself an eager compan-ion and Mittag-Leffl er went on many a walk together with him and Sonja. One day they climbed de Brocken, the highest mountain in the Harz and stayed overnight there. The next day, when they returned to Möllers Hotel, they found out that Adolf Hurwitz had arrived. The next few days were fi lled with what Mittag-Leffl er referred to as talks and conferences – mostly with Weierstrass – as well as walks with Cantor, Sonja and Hurwitz. Mittag-Leffl er found Hurwitz to be a modest young man fi lled with new and good ideas, with a genuine love for his sci-ence. ‘Sonja has mobilized her full battery of coquettish charms aimed at his simple innocence,’ he added.

At the dinner table on 16 August, Sonja started a dis-cussion concerning the liberation of women, something which would turn out to be very unpleasant for Mittag-Leffl er. Present at the table, in addition to Sonja and Mit-tag-Leffl er, were Hurwitz, Cantor and the two sisters of Weierstrass; Weierstrass himself never had his dinners there. Mittag-Leffl er, according to his diary notes, did nothing but develop his own ideas that women should also have the right to vote, when Sonja suddenly inter-rupted him and started to talk about how much more than

him she did and achieved in Stockholm. She lectured twice as much and had writ-ten several treatises; she had never had a leave of absence and had never been sick. Mittag-Leffl er responded calmly that he did not want to defend himself but noted in his diary that he had been on the verge of remarking that included among his du-ties in Stockholm was all the work involved in maintain-

ing the position for his female colleague. Cantor on the other hand reacted to Sonja’s outburst by pointing out that she had forgotten all the work Mittag-Leffl er did with Acta Mathematica. But this was sprightly dismissed by Sonja, who claimed that the work with the journal did not make more demands on his time than did her own domestic chores on hers, and besides he had a secretary.

A bit later in the day he realized that Sonja had also spoken disparagingly of him to Weierstrass. And from Weierstrass he learned that Schwarz, who was due in Wernigerode two days later, thought of Mittag-Leffl er as his adversary. The reason was supposedly that everyone believed that Mittag-Leffl er was fi shing for a position in Berlin, an ambition, he insisted in his diary, that had not been admitted to his thoughts, although it certainly would be worth thinking over. His most fervent desire for the coming winter was only to get enough energy to work with mathematics, he noted, and concluded discouraged at the end of that day: ‘But it will not be so. Things are just getting worse and worse and it will probably soon end as it did for my father.1

Through daily meetings with Weierstrass the assess-ment of the submitted treatises progressed. The work of Poincaré was in particular discussed and Mittag-Leffl er was continually struck by the penetrating acuity of Weierstrass’ mind, his unusual powers of recollection and how easily he quenched even the most diffi cult of questions.

Schwarz’ presence in Wernigerode led to several un-pleasant scenes. Both Sonja and Weierstrass had writ-ten to Schwarz and asked him to come and take part in constructing a model for the body of rotation with which she was working for her prize-winning essay. In addi-tion Weierstrass had asked Schwartz to come in order to try to bring about reconciliation between him and Mittag-Leffl er. At the fi rst meeting at the dinner table, Schwarz bowed to him several times and Mittag-Leffl er thought that he had looked very embarrassed. Later in the evening, however, after Schwarz had retired and just Sonja and Cantor remained, Mittag-Leffl er was for the fi rst time told about the details of the rumours Schwarz had spread about him during the last few years and he

Georg Cantor (1845–1918) © Institut Mittag-Leffl er

Adolf Hurwitz (1859–1919) © Institut Mittag-Leffl er

Herman Schwarz (1843–1921)

1 Mittag-Leffl er’s father had a mental breakdown and was committed for the rest of his life to an institution, while Mit-tag-Leffl er was still in his youth [translator’s note]

History


started to grasp what the persistent strain in his relations with Schwarz was really founded on. Everywhere in Ger-many, France and Italy, Schwarz had disseminated the ru-mour that Mittag-Leffl er was a lecher, a Wüstling, leading a life of debauchery – so morally depraved that he had married Signe while he was still suffering from syphilis. That was why she had not been able to give birth and that was why she had consulted doctors all over Europe. To substantiate his claims he had bragged that he had sought out Mittag-Leffl er’s doctor in Helsinki. Cantor confessed that after having heard this he had refused to shake hands with Schwarz and that he would rather not have anything at all to do with such a person. And when Schwarz had spread the same rumours in Paris, Poincaré had reacted very strongly and forbidden anyone to bad-mouth his friend Mittag-Leffl er and that the rumours, even if there were some substance to them, were the sole business of Mittag-Leffl er himself and no one else. Both Poincaré and Picard had threatened to have him evicted out of the country if he would not give up his slanderous allegations.

Cantor continued the next day to reveal more about Schwarz and how the story about the syphilis had been expanded upon. Cantor was curious as to what steps Mit-tag-Leffl er would take. Cantor was also able to relate that when Schwarz was newly engaged to Kummer’s daugh-ter, he had asked for advice about whether he should tell his future father-in-law that as a soldier in 1866 he had had a severe episode of syphilis.

‘Did not sleep during the night,’ Mittag-Leffl er noted in his diary. He had decided to confront Schwarz with what he had learned. At the dinner table on 20 August he asked Schwarz for a meeting between four eyes and just after dinner he escorted Schwarz to his room. Mit-tag-Leffl er started by remarking that he had expected an unconditional apology. When that did not come about, he minced no words in telling him that he could expect something very unpleasant. In Germany such defama-tions would be punished and so many witnesses could be brought up from Germany, France and Italy that there should be no doubt as to the outcome. He would be condemned to at least six months in some correctional institution, in addition to being saddled with paying in-demnities. According to Mittag-Leffl er, Schwarz only stuttered something about having come to Wernigerode only because Weierstrass had asked him to and that Mit-tag-Leffl er was to do what he thought just. Much more was not said as Schwarz was then asked to leave.

Mittag-Leffl er went right away to Weierstrass to tell him what had transpired but as he found him asleep, he instead sought out Sonja. She showed indignation as to the behaviour of Schwarz but thought nevertheless that Mittag-Leffl er should not have brought about such a scene. She meant that Mittag-Leffl er had only thought of himself, not on how the affair would affect Weierstrass. He countered that he had not thought of himself but of his wife Signe. Had it not been for the lies spread about her, he would have done nothing. Sonja was neverthe-less not fully satisfi ed, something Mittag-Leffl er related to her need of Schwarz for help in her own work.

When Weierstrass was later fi lled in on what had happened, he thought that Mittag-Leffl er had acted in the right way, even if he found the whole thing very em-barrassing and painful. He may not have believed that Mittag-Leffl er would be satisfi ed with what had already been said and he took it upon himself to force Schwarz to give a full apology. In his diary Mittag-Leffl er noted that since a Swede could not duel – and since he had now upbraided Schwarz so strongly, was it not rather Schwarz who could demand satisfaction? – there remained noth-ing but a legal process. But he did not look forward to bringing about such a scandal.

The same evening Weierstrass succeeded in extracting an apology from Schwarz and his word of honour that he had not spread such stories during the last few years and that he would never ever do it again. Mittag-Leffl er who desired to accommodate Weierstrass decided to let mat-ters rest for his sake and declared that he accepted the apology. Cantor, on the other hand, thought that Mittag-Leffl er ought to have demanded an apology in writing, something Mittag-Leffl er admitted he would have done had it not been for Weierstrass – anyway he would ‘treat Schwarz as air’ until he came around and personally asked for forgiveness.

The next day Mittag-Leffl er noted in his diary that he slept extremely well during the night. During the after-noon Weierstrass told him that he had had another talk with Schwarz, who in the presence of Weierstrass would like to ask for forgiveness. Schwarz once again swore that during the last year, he – after Weierstrass had admon-ished him during the Easter of 1887 – had not told any-one the story, whose veracity he no longer believed in. But he had heard the story from the Finn Neovius, who had relayed the contents of a Finnish short story in which this had been told and had been informed that Mittag-Leffl er had been the model for it.

In the evening Weierstrass took the two of them for a walk but when, after the evening meal, he tried to bring about a reconciliation, Schwarz excused himself saying he was tired and wanted to retire to bed. Not until the next afternoon did Schwarz arrive along with Weierstrass and present his apology for what had happened. Mittag-Leffl er promised to forget all about it and hoped that in the future they could work without hostility and with a mutual desire toward common scientifi c goals. He ex-pressed a certain understanding that because of the strong competition between mathematicians in Germa-ny, there could arise such back-stabbing but that he, who was above all such things, ought to be allowed to live in peace with everybody else. Schwarz started then to claim that he did not begrudge anyone nor compete with any-one but was stopped by Weierstrass who said he should not talk about things that were not true.

During a walk in which everybody took part Schwarz expressed his intention to give a talk, which he later wanted to write down and send to Acta as a proof for all the world to see of their reconciliation. Next afternoon Schwarz did give a lecture but it seems never to have been sent to Stockholm.

The next days in Wernigerode were devoted to further

History


discussions of the manuscripts for the prize of King Oscar II and both Schwarz and Hettner participated actively in these. In addition Mittag-Leffl er and Weierstrass tried to help Sonja in integrating a differential equation that had arisen in connection with her problem. Weierstrass was also working on the stability of the n-body problem.

A kind of serene sense of consensus seemed to have descended on them those last days at the Müllers Hotel. When Schwarz left on 15 August, he showered Mittag-Leffl er with declarations of friendship. The only things that created some discord and kept confl icts alive were the machinations of Sonja. And Mittag-Leffl er thought that the reason that Sonja had been so unpleasant was her jealousy of the kindness Weierstrass had shown him. Many times their old teacher had turned to Mittag-Lef-fl er before he had addressed himself to her. Now she did not want to hear anything about Mittag-Leffl er’s plans to invite him to Sweden next summer. As she was not going to be in Sweden at the time, there was no reason why Weierstrass should be either.

Weierstrass told Mittag-Leffl er that he strongly disap-proved of Sonja’s plans to get a position in Paris and he resented her lack of tact in that she exploited every op-portunity to speak disparagingly of Stockholm. He had always reminded her that no other country would have done as much for her as Sweden. On 26 August, Mittag-Leffl er concludes his diary with the following.

My decision is made; I will try to bring it to the point that she [Sonja] will be if not personally indifferent to

me, at least far more distant than up to now. Only then can I endure all those pains that she infl icts on me. So long as she is so personally close to me, as she has been up to now, she tortures the life out of me. I can put up with much from those whom I am indifferent to but nothing at all from the few who are really close to me.

On one of the last evenings, Mittag-Leffl er and Cantor went down to the centre of Wernigerode, where a large crowd of people had congregated and where festivities had been arranged in connection with the silver jubilee of the wedding of the local prince Graf Otto zu Stolberg-Wernigerode. The bridal couple, accompanied by serv-ants, were driven through the town in elegant wagons towards the castle. ‘The whole made a tragicomic impres-sion,’ Mittag-Leffl er wrote in his diary. On 28 August, he left Wernigerode.

Arild Stubhaug is an established liter-ary writer in his native Norway. Among mathematicians, he is known world-wide for his biographies of Niels-Henrik Abel

and Sophus Lie. Both books have been translated to Eng-lish, French, German and Japanese; the fi rst metioned one, moreover, to Russian and Chinese.

IRMA Lectures in Mathematics and Theoretical Physics Vol. 12

Quantum GroupsBenjamin Enriquez (IRMA, Strasbourg, France), EditorISBN 978-3-03719-047-0June 2008, 141 pages, softcover, 17 x 24 cm38.00 Euro

The volume starts with a lecture course by P. Etingof on tensor cat-egories (notes by D. Calaque). This course is an introduction to tensor categories, leading to topics of recent research such as realizability of fusion rings, Ocneanu rigidity, module categories, weak Hopf alge-bras, Morita theory for tensor categories, lifting theory, categorical dimensions, Frobenius–Perron dimensions, and the classifi cation of tensor categories.

The remainder of the book consists of three detailed expositions on associators and the Vassiliev invariants of knots, classical and quan-tum integrable systems and elliptic algebras, and the groups of alge-bra automorphisms of quantum groups. The preface sets the results presented in perspective.

Directed at research mathematicians and theoretical physicists as well as graduate students, the volume gives an overview of the on-going research in the domain of quantum groups, an important sub-ject of current mathematical physics.

ESI Lectures in Mathematics and Physics

Recent Developments inPseudo-Riemannian GeometryD. V. Alekseevsky (University of Edinburgh, UK) and H. Baum (Humboldt-Universität, Berlin, Germany), EditorsISBN 978-3-03719-051-7. June 2008, 549 pages, softcover, 17 x 24 cm. 58.00 Euro

This book provides an introduction to and survey of recent develop-ments in pseudo-Riemannian geometry, including applications in math-ematical physics, by leading experts in the fi eld. Topics covered are:

Classifi cation of pseudo-Riemannian symmetric spaces, holonomy groups of Lorentzian and pseudo-Riemannian manifolds, hypersym-plectic manifolds, anti-self-dual conformal structures in neutral sig-nature and integrable systems, neutral Kähler surfaces and geomet-ric optics, geometry and dynamics of the Einstein universe, essential conformal structures and conformal transformations in pseudo-Rie-mannian geometry, the causal hierarchy of spacetimes, geodesics in pseudo-Riemannian manifolds, Lorentzian symmetric spaces in su-pergravity, generalized geometries in supergravity, Einstein metrics with Killing leaves.

The book is addressed to advanced students as well as to researchers in differential geometry, global analysis, general relativity and string theory. It shows essential differences between the geometry on manifolds with positive defi nite metrics and on those with indefi nite metrics, and highlights the interesting new geometric phenomena, which naturally arise in the indefi nite metric case. The reader fi nds a description of the present state of art in the fi eld as well as open problems, which can stimulate further research.

European Mathematical Society Publishing HouseSeminar for Applied Mathematics, ETH-Zentrum FLI C4, Fliederstrasse 23CH-8092 Zürich, Switzerland [email protected]

Interview


Interview with Preda MihailescuGöttingen (Germany)

Preda V. Mihăilescu, born in Bucharest, Romania, is best known for his proof of Catalan’s conjecture. After leav-ing Romania in 1973, he settled in Switzerland. He studied mathematics and informatics in Zürich and then pursued a career as a mathematician in the Swiss banking sector. While still working in industry, he received his PhD from ETH Zürich in 1997. His thesis, titled Cyclotomy of rings and primality testing, was written under the direction of Erwin Engeler and Hendrik Lenstra.

Mihăilescu started to work in academia as a research professor at the University of Paderborn, Germany. Since 2005 he has been a professor at the Georg-August Univer-sity of Göttingen (Germany).

This interview took place on 3 December 2007 at the Mathematisches Institut of the University of Göttingen. It was prepared by Axel Munk (Mu) and Samuel Patterson (Pa), both from the University of Göttingen, who also par-ticipated; it was conducted by Hanno Ehrler from Deut-schlandfunk, a German public broadcasting channel.

Childhood and education, Romania and Switzerland

First of all, I would like to ask you to tell us a little about your curriculum vitae, starting with your child-hood. I read in an article that you had a talent for mathematics and numbers very early on.The game with numbers indeed started about the age of four, and before I was fi ve I had the reputation of the child who multiplies three digit numbers. I liked the game and could not understand what others found so as-tonishing about it – a little bit like listening to music that not many appear to notice. There were variations; friends of the family who were engineers visited and verifi ed my 237 × 523, say, with a slide rule – and I found it quite awk-ward when I looked at “the competition”. I saw all these lines at varying distances on the slide ruler and realized that it could never yield the precise 6 digits of the result, since the maximum accuracy was 5 digits. So I knew that the verifi cation was only approximate – yet I knew that my result was precise. It was different on the playground;

only much later did I learn that the older people whom I did not know and who came making their challenges – sometimes they even tried to teach me to perform mul-tiplications on paper – had actually made bets with their friends.

Of course this gave a predisposition for mathematics, but it was not cultivated in any particular way for quite a while. Humanities came fi rst in my early life.

What other important recollections do you have from your childhood?My childhood was relatively protected at a time of in-tense dictatorship. My parents did not hide from us all of what was happening. Thus we knew that an uncle was in a camp and that others had no job or worked as street dog catchers (it happened), having been poets and philoso-phers before that. We knew that twice a year my grand-father visited his former professor and master, who had recently fi nished his camp sentence, taking him a large dried sausage, which made the old man cry. Then they had several hours of discussions, which brought them back into the spiritual fi elds of their youth; I sensed a particular intensity in my grandfather when he was com-ing back from these visits, like a veil was raised for a brief period of time over a world I would never encounter in my own life.

We knew that we had to be careful about what we said and we feared the power – but in fact I had little di-rect contact with “those guys” as a child. The parades, the songs and the doctrines were visible but not so tangible as the family circle that was offering comfort. Eventu-ally, when going to school, a third category emerged: the school teachers, some of whom were really nice – yet they were employed by “them” so one could not be sure. For us, Stalinization came to an end towards the mid-60s, al-though it formally ended in 1956. In the 60s there was, on the one hand, an amnesty and many people came out of camps, and on the other hand … there was “Strawberry fi elds” and “Penny Lane”. If I recall correctly, we heard the fi rst Beatles songs even on public radio. By the time I was able to consciously ask the question: ‘why did one grandfather die under untold conditions while the other one lives with us to the greater joy of all, why did uncles disappear, and whether or not we had been close to such a destiny?’ a sort of a warmer wind was blowing.

And then, when I became a teenager I was elected, as a good pupil, to some leading communist youth role in the school. I thought I was choosing a smart way out by basically saying at the meetings that one should try to be oneself even within this organization because accord-ing to what they said, they actually expected this from the youth – and not, like everybody knew, frightful obe-dience and indoctrination. The words were not exactly these but the meaning was quite clear, and after several

from left to right: Samuel J. Patterson, Preda V. Mihăilescu, Axel Munk and Hanno Ehrler

Interview


calls to order in offi ces of higher ranking “comrades”, I came to understand that I did not have all one needs for civil disobedience. This may be one of the personal ex-periences that made me choose, four years later, to seek asylum in Switzerland.

And there you studied mathematics?And there I studied mathematics and entered a new life that was challenging in all directions.

There were two things that led me to the choice of doing applied mathematics: one was the pragmatic esti-mate of survival odds for the inexperienced refugee that I was. But this could never suffi ce, not without the feeling of being in a world where one could do things, a lack of limitations that had to be experienced. So I went to do something that none of the males of my family had done, at least in the last three generations – become practical. They had all ended up teaching in one way or another; I was going to work in some productive area – quite ironic, looking back now, isn’t it? It is probably from my mother that I have learned a love for very practical, well done things. She was a gynaecologist and a surgeon. Delivering babies under the most diffi cult conditions, both medical and physical, was an unspoken, penetrating passion for her. From her I think I received the love for things with a practical purpose – developing a project like delivering a child, bringing something new to life…

Industrial career

So it is quite natural that you studied applied math-ematics and then went into industry. What did you do there? What kind of work did you have to do?There are two periods of my industrial career. After the second “Vordiplom” in mathematics until the end of my computer science Masters degree, I worked, with interruptions, in a company in the machine industry. I applied numerical analysis and developed methods and programs for designing models of turbine blades so that the gas fl ow around these blades could be computed exactly. It is a typical application meant to provide ref-erence data for testing larger software packages. From the desire for a visual interaction with these models, I was led to several visualisation applications; in the end, I even developed a visual interface for software that computes load fl ow and short-cut simulations in large electrical networks. It was an effective sales argument for that specialized, professional software and my fi rst application that had an impact on the market. That was also in the very early beginnings of graphical interfaces so I had to invent many of the tools I needed pretty much from scratch.

Then, after completing my computer science studies, I switched to information security. I worked in that do-main fi rst in banks and later in a consulting and develop-ment company. I would say that in the productive phases of my industry life, I was peacefully conceiving projects and bringing them to life, and there was nothing spectac-ular about it. The security of online ATM system in Swit-zerland, for example, which has been running since 1990

with some natural renewals and no essential changes to the core, has withstood the test of time and represents the silent success behind that work.

What does “information security” mean? Were you concerned with security systems?Yes – security of information. That means protecting data on the net against intrusion or modifi cation, protecting identities – in the sense that the identity of a person may be associated with documents or “electronic signatures” in an inalterable and irrevocable way – and variations of these basic problems of confi dentiality and identity. The methods used belong to the fi eld of cryptography, which uses number theory, and this is why a mathematician was preferred. The mathematical background was certainly useful so that the cryptographic issues were easily ac-cessible. From my point of view – and probably not only mine – the challenge of the job was to supervise a large system so that no security gaps arose. Gaps due to or-ganizational reasons rather than purely cryptographic ones often played the most important role; cryptography is well under control due to the theory.

It was also the beginning of public key cryptography and I had a lot to do helping potential users be a bit more sensible. Primarily, security made the life of the normal programmer or computer user slightly harder, not only because of the numerous password protected applica-tions one had to sign into sometimes dozens of times a day but also because a secured application could, at that time, be logically slower. One had to convince co-work-ers to accept such inconveniences to enforce the security of the internal network.

At one stage, I was employed in the most important bank in Switzerland, which had the ambition of being ahead of its time in IT; it could also afford to pursue this goal. At that time the term “single sign-on” was starting to be coined as a desirable alternative to the multiple applications asking for a password on the desk of every employee. The technology answering the expectations of this time was only developed over fi ve years later.

I was asked if I “could develop a single sign-on sys-tem” for the company’s network. What I was doing practically at that time was cryptographically securing the communications in the bank. This was very useful and could be done with the manpower available. I was unsuccessful in using a detailed technical argument to explain that “single sign-on” was not at that time with-in reach. On that occasion, I learned that sometimes a negative proof is not accepted as a basis for a decision. Maybe that experience made me wish for an environ-ment where a negative proof is well regarded; anyhow, years later I was able to present a negative proof, which confi rmed Catalan’s conjecture, and that one was very well accepted.

Pure mathematics during spare time…

So we are back to pure mathematics. As I know, you also had time for studying pure mathematics; after all, you solved Catalan’s conjecture. How was it possible to

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have this job in the bank and the work in industry and at the same time be concerned with pure mathematics? Is it some kind of hobby or an inner drive?The inner drive was there and mathematics was also help-ful for my mental balance. For instance, I used to work at the weekends or in the evenings at or around my PhD. And the PhD was about…The PhD was about primality testing and was thus in ap-plied number theory with an explicit application to cryp-tography too. It was completed during my time at the bank. I then liked designing effi cient algorithms. Some of the ideas from that time came to fruition later, for in-stance recently in the fastest general primality test for large numbers.

…and as a main occupation

But at some time you decided to move from your indus-try job into research. Why did you make this decision?Probably the feeling had accumulated that I wanted to do more mathematics. It happened that I was offered a position at the RSA labs, the research group of a ma-jor cryptography company in the USA, and, at the same time, at a university. Doors were open in both directions; in fact, I fi rst went to RSA for several months and after that I accepted the university position, at Paderborn Uni-versity in Germany. I guess my inner voice told me that I wanted to do academic research and teaching too. In Boston, I was already working at Catalan in the evenings – one may speculate whether this had an impact on my decision too … it is probably just speculation!

Pa: Did the university come to you or did you apply for a university position at this time?I corresponded with Professor von zur Gathen (Pader-born University, Germany), who had invited me before to give some seminar lectures on the results of my PhD. We kept in touch after that and in fact he offered me a position that was also connected to some cryptographic applications, yes.

Catalan’s conjecture1

Let’s speak shortly about Catalan’s conjecture. How did it come that you were concerned with this particu-lar problem and how did you solve it?This is more than one question…

Maybe you could tell us how you got involved with the problem and on which scientifi c shoulders your solu-tion was built upon.Let’s try to break this down into several questions. Firstly, why did I get concerned with Catalan? I came across it at a conference in Rome during an interesting talk by Guil-laume Hanrot. Returning to Paris where I was spending a summer doing research, I found myself alone and I worked several days to understand more about Hanrot’s methods. I did not stop until I had more than a proof of that conjecture! It happened that within a week or two, I

could do a step that was then considered to be non-neg-ligible; I proved the so-called “double Wieferich condi-tions”. That was encouraging and may have strengthened my tenacity. Then I was in Paderborn, where I had the peace of mind to think over a longer period about this problem – it was different from the evenings of math-ematics after work.

You were asking about the “shoulders” on which I was standing. I would at fi rst answer that I was jumping from shoulders to shoulders. At the beginning it was the shoulders of various people who had investigated Cata-lan’s conjecture in the last 30 years; there’s a long list. After Bugeaud and Hanrot, who gave me a taste for the question, I got acquainted with Maurice Mignotte, who was very active in the area. I had to understand the ideas of Inkeri and Schwarz, and Bennet, Glass and Steiner, Tijdeman, and, of course, the consequences of the work of Baker. At the base of all practical results there was an analytic simplifi cation made by Cassels in the 1950s – everybody needed that result as a starting point. I cer-tainly forget half of the important names… However, today I would claim that the shoulders that still hold the present proof are those of Kummer, Leopoldt and Thaine, who gave the fundamental facts from cyclotomy that are used.

Perspectives of number theory

Catalan’s conjecture is in the fi eld of number theory. What are the perspectives of number theory today?Pa: I’m glad this question has been asked of you. (smiles)Mi: Glorious as ever … I am sorry; on the one hand, I do not retain the authority to expand on this question – there are much more competent people to do that. And I can imagine they would also be very careful with prog-nostics. But since I stand here as one who has worked applying mathematics and who has also done research, I would like to recall one impressive example. Algebraic geometry in positive characteristic was born with Weil and Grothendieck (pursuing work done by Dedekind and Weber, and Kronecker and Artin) as I see it – being the amateur expert that I am – in the second half of the last century. It took some time to be established among geometers themselves. In the 80s however, with Schoof’s algorithm for fast counting of points on elliptic curves defi ned over (large) fi nite fi elds, the topic became in a short time a domain of intensive research in algorithmic number theory with practical applications in cryptogra-phy. It is hard to say where such phenomena happen or predict how they will happen again in the future, but they certainly will!

1 Mihăilescu’s theorem (formerly Catalan’s conjecture) was conjectured by the mathematician Eugène Charles Catalan in 1844 and proved in 2002 by Preda Mihăilescu. It states that the only solution of the equation xa – yb = 1 for x, a, y, b > 1 is x = 3, a = 2, y = 2, b = 3. The proof appeared under the title Primary cyclotomic units and a proof of Catalan’s conjecture, J. Reine Angew. Math. 572 (2004), 167–195.

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The Göttingen tradition

We are here at the University of Göttingen and, as Pro-fessor Munk told me, there’s a tradition at this univer-sity of combining pure and applied mathematics. Pro-fessor Patterson, could you tell us a little bit about the tradition of this university and about the state of art today.Pa: Well, the origins of mathematics in Göttingen really have to be traced back to Carl Friedrich Gauss, who was the director of the Sternwarte (the observatory here) for a very long period of time. He wasn’t actually a professor of mathematics – he was a professor of astronomy – but he was regarded as the leading mathematician in Europe at the same time. So at this point you have someone who was, for example, doing very practical work during the tri-angulation of the Kingdom of Hannover, and on the other hand developing, using this experience, the abstract area of differential geometry. And the tradition of Gauss has essentially inspired everybody in Göttingen since then, in one way or another. One of the people who took this up most strongly was Felix Klein, who was very, very much involved in the connections between mathematics and in-

dustry. He didn’t get much gratitude. Such questions were hotly debated about 110 years ago. Klein was involved in a whole series of public arguments. This was part of the development of the Prussian state and industry.

And at present? Is there some kind of tradition that leads to the present in the university here?Pa: Well, we are like most universities. We are quite small as a university, as far as mathematics is concerned, but there are three mathematical institutes inside the faculty. One is called, for historical reasons, the Mathematisches Institut, which is merely pure mathematics now, and then there are the two applied mathematics departments: the Institutes for Numerical and Applied Mathematics and the Institute for Mathematical Stochastics. If I may allow myself a personal opinion here, I would like to see, within a reasonable amount of time, all of these under one roof and working together with one another. At the moment this is not about to happen and the present planning is to move to a new building in about nine years.

The perception of mathematics in industry

In your biography you had both: on the one hand, you worked in applied mathematics in industry and on the other hand, you were working in research and now in academia. I would be interested to hear a little bit about how industry views mathematics? Let me give some partial answers and perspectives. First, there are those domains that are traditional “customers” of mathematics: mechanical and electrical engineering, fi nancial and insurance mathematics and a few more. There, mathematics is probably supposed to be an impor-tant tool for the precision and accuracy of the required application and the image of what one can expect from mathematics generally has precise contours. This is sim-ply due to tradition.

And outside these traditional branches?There have been many more recent applications born with the development of the computer. From statistics and information theory to the very dynamical branches of computer science – they are encountered almost eve-rywhere, from the pharmaceutical industry to transporta-tion scheduling, from the food industry to, say, fl ood and earthquake prediction.

And what is the reception of mathematics here?Frankly, I would not know the answer. An educated guess may be that every domain reaches a degree of complexity beyond which the immediate solutions are not suffi cient and some mathematical understanding is called for.

And this would then have a positive impact on the im-age of mathematics!?In principle, yes, depending on how widespread the awareness is that a change was brought about by math-ematics. After all, maybe to a large extent, people in in-dustry care less about the distinctions between various scientifi c disciplines, as long as things work… Therefore,

New buildings under construction for Institut für mathematische Stochastik

Mathematisches Institut Göttingen at the times of Courant and Hilbert and today

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there is a difference between the image of mathemat-ics and the image of mathematicians. Maybe the latter is more distinct and I believe they are, in general, well regarded for their ability to structure complex problems, identify possible solutions and modify targets so that they become affordable.

You mean that mathematicians are asked for in industry?I mean they are well regarded. There is often the principle of learning by doing in industry. You may fi nd a physicist solving integer programming problems or a number the-orist becoming a software manager or a bank manager. So, mathematicians have a good reputation for coming around to problems on which they are trained and those for which they are not. They soon have to discover that they belong to a micro-culture to which absence of proof is close to illusion. This is a positive contribution to the collective, especially when they do not expect that the others adhere to the same standards of formal consisten-cy. It is a good habit to explain convictions, even obtained by some proof reasoning, in terms closer to common lan-guage. Most people may resent an excess of precision and accuracy of expression as a kind of snobbery. How did you experience the change from industry to academia?The time frames are certainly different in industry and it is sometimes hard to pursue an important idea over a longer period of time. My taste and my own history make me long for places where the two meet or it is fore-seeable that they could meet. To an extent, I was offered this opportunity in Göttingen for this reason. Here it is possible to pursue both theoretical research in number theory at the Mathematical Institute and practical re-search in biometrics at the Institute for Mathematical Stochastics.

Biometrics

Mr. Munk, in your institute you work jointly with Mr. Mihăilescu on biometrics. Are there many groups in mathematics that are working on biometrics?Mu: No – interestingly, this is not the case. Worldwide, there are many groups, either in academia or in industry, who work on that issue, of course, but usually their back-ground is from computer science, in particular pattern recognition or electrical engineering. The group here is, as far as I know, the only larger group in a mathemati-cal department that deals extensively with those issues – and that certainly makes it unique. Indeed, in our group we bring together experts from several areas. Preda Mihăilescu has a background in cryptography, on the one hand, and on the other hand, in biometrical identifi ca-tion analysis. We combine his expertise with my statisti-cal knowledge and our knowledge in pattern recognition, image processing and enhancement and specifi c aspects of geometry. Actually, our research process itself gradu-ally increased the insight that many interesting mathe-matical questions are inherent to biometric issues.Mi: There is a nice historical analogue to what we are

doing. At the beginnings of data transmission with mo-dems, the data were transmitted in clear text and the transmission rate was low. There is a fundamental law of information theory, Shannon’s law, that states that one cannot transmit data over a channel at a rate beyond the so called “channel capacity”, which is essentially limited by the physical channel used: wires, radio, etc. That physi-cal limitation was hard to relax, so nobody knew how to reach better transmission rates for modems. Some peo-ple believed this was not possible without improving the physical channels – and that belief was scientifi cally backed up by Shannon’s law. Yet, Liv and Zempel looked at the problem from a different perspective and found that one does not have to send more text at a time in order to increase the rate. It suffi ces to increase the infor-mation that the text contains – and for this, one can com-press the text before sending it on the physical channel and then decompress it, since in fact most of the data sent over the net have a high rate of redundancy! By using compression, the transmission rate suddenly increased, without changing the physical channels and without con-tradicting Shannon’s law.

Fingerprints

Mi: In fi ngerprint security we are in a somewhat simi-lar situation now. Our group has proved that the security based on the currently used fi ngerprint data is bounded and in fact insuffi cient. Using presently extracted data, one can – by state of the art methods – never reach the degree of security required for internet transactions. The proof is based on statistical evidence from the literature, evidence we want to back up by our own extended fi eld research. But the order of magnitude is quite reliable. So, by changing the perspective, as Ziv and Lempel did, one is led to the observation that the fi nger contains a wealth of information that is currently not used in AFIS (Automated Fingerprint Identifi cation Systems). The hu-man operator uses this passive information, so she can sort out evidence in cases in which the machine does not. Obviously, we need new algorithms capable of extracting and structuring such information; this will also increase the security rate. This was one major goal for our group from the start. We fi rst thought only of improving the identifi cation rate – now it shows that the same work can help improve the security systems based on biometry.

I think it would be interesting for the readers to de-scribe the different credentials that are present in the Institute for Mathematical Stochastics; moreover, what are people actually working on?Mu: In fact, we combine the expertise of various peo-ple with different backgrounds. As I said before, Preda Mihăilescu has an expertise in cryptography and practical biometrics. Then we have somebody whose background is from differential geometry, obviously an area of pure mathematics, that turns out to play a very important role in the modern analysis of fi ngerprints. My own background is statistics; we have people with a degree in computer science and even a degree in both computer science and

Interview


mathematics. We collaborate with people from physics and from biology. This makes such an area more attractive for me personally; it is interdisciplinary in the true sense.

At the end of the day, I think mathematical model-ling plays a very important role, and this can and has to be done by mathematicians. This is probably the particu-lar strength of our group: we tackle various applications from the perspective of mathematics. Our major focus is mathematical modelling and the understanding of math-ematical structures that we use for fi ngerprint identifi -cation, for example. This is then combined with statisti-cal issues, e.g. the investigation of the distribution of the main characteristics of a fi ngerprint on the fi nger. Just view your own fi nger tip and you experience a nice fl ow fi eld of ridges and bifurcations with fascinating random patterns – a living geometry!

Human vs. machine expertise

Talking about biometrics, where do you see the major challenges in the near future where mathematical ideas will make the difference?Mu: An important issue, which I think will play a major role in the next few years in the whole area, is the in-tersection between biometric identifi cation systems and security issues, i.e. security issues in the sense of cryptog-raphy. Here, Preda will certainly play a prominent role in our group. Furthermore, statistical ideas are required. Traditional cryptography uses reproducible information – any password or secret key is a unique chain of char-acters from an alphabet. In contrast, fi ngerprints have an inherent variability that has to be taken into account and cannot be avoided. This leads to a paradoxical situation: security under variability of the key. This seriously lim-its the potential security of biometrical systems! It ap-pears that the state of the art approaches are unaware of this limitation in various applications, in particular in ‘soft biometrics’, where time and money often imposes restrictions to security. However, I believe that a com-bination of thorough statistical understanding of these limitations and innovative use of additional fi ngerprint information, which so far has been neglected as well, will

improve both reliability and security of such systems – a great challenge for the future! It is by thorough under-standing of the nature of limitations that one can, like in the case of Liv and Zempel, fi nd a way to bypass them.

Mi: I’d like to add that this is a domain of an intellectual nature where there is some unknown mathematics. It is interesting that computer scientists who have worked in “expert systems” have repeatedly faced the problem of learning from human experts. Some simple general rules can be easily understood and transformed into algo-rithms but then running “man against machine” on some specifi c decision problem, one fi nds that human experts tend to be better, yet they cannot explain to the computer scientist the choices behind their decision processes. This is a complex problem and certainly the human has a wide capacity to integrate – let me use the word holistically – complex, even apparently irrelevant data, and there-fore improve the decision. We are thus trying to improve the algorithmic integration of the layers of information by learning from humans. If we make some step ahead in this direction, certainly the immense processing capac-ity of the computer will then make it possible to use the machine for double-checking expert decisions in critical contexts; experts do make errors too!

Mathematics has the capacity of modelling. Situ-ations like this one teach us a certain kind of humility, since the major, powerful tools of mathematics don’t get to the facts. There is a need for some interaction with the object that approaches a mathematical and machine understanding of what man was doing before. We are in this process, and it brings up new notions and new ideas, which then offer a feedback to the possibility of bringing more well-known – or possibly new – domains of math-ematics into the game. We are at this turning point.

Can you give an example of what man did before the machine?Yes, certainly. If you look at fi ngerprints, it was defi ned in a somehow conventional way that the so called minutiae identify the fi ngerprint and thus the person. Minutiae are, for instance, line endings and line bifurcations on the fi n-ger. Matching suffi ciently many of them (12–18, say, ac-cording to domestic laws of various countries) identifi es a person in court. However, an expert looks at the whole image of the fi nger and he uses everything he sees there for the orientation and thus for the matching of minutiae. The machine was (essentially) only taught to use the ac-tual location of minutiae. So the integrative process will probably need to take ridge connections, curvatures and further visual impressions into account, which the expert may use in case of uncertainty.

Mu: But there is another aspect, and maybe this clarifi es why we require this broad range of expertise: from sta-tistics, imaging, pattern recognition and cryptography. If you, for example, simply raise the question of how secure a fi ngerprint identifi cation system can potentially be, various issues have to be considered. One is the pattern recognition part, which means how accurate the informa-

Screenshot from fi ngerprint development and analysis software

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tion is that the algorithms extract in order to optimize the identifi cation rate. The other issue, and no machine can answer this question, is how good the quality of fi n-gerprints are in nature in order to make these things work, i.e. what are the inherent natural limitations of the achievable error rates in biometrics. This is not very well understood, albeit of great practical relevance.

For example, we have recently started a project with the Bundeskriminalamt – the German Criminology Agen-cy – where we investigate statistically how fi ngerprints transform over a lifetime. There appears to be a problem in quality with prints of elderly people and very young peo-ple. Fingerprints change during relatively short time peri-ods in these age groups and this is not understood at all.

This causes diffi culties for the minutiae – based ap-proach, which is used in AFIS nowadays, because there is some physical distortion of the fi nger, especially with teenagers. It is not easy to identify the same person over a gap of two to three years because of fi nger growth. But you can do it if you use the background information that we are extracting, since the line connections are still the same; it is just the rectangular reference grid of the minu-tiae that is distorted, like one of those popular paintings of Vasarely.

Pa: I’m just wondering here – fi ngerprints have been used for 100 years. Was this not a problem before now?Mi: As I explained, the human expert had less of this problem. On the other hand, the human can never proc-ess the huge amount of data that a machine can. We wish to teach the machine to use more information in uncer-tain cases and thus approach the differentiated attitude of the human, while keeping its specifi c high performance.

Mu: And there is another issue that is very important. At the end of the day, the target of your identifi cation sys-tem is to guarantee certain error rates, irrespective of the particular method used or even of the fact that human experts are involved or not. However, the requirements set for these error rates logically depend on the applica-tion. Thirty years ago, fi ngerprints were mainly used in forensics but nowadays we are talking about fi ngerprint identifi cation in a variety of applications, including com-mercial systems like access control to personal comput-ers or cell phones, where the security does not need to reach forensic standards. In contrast, other commercial applications involving fi nancial transactions require very low error rates and are only just developing.

Security issues

A particular problem of your research is probably to estimate the security of biometrical systems?Mu: How secure is the method? Certainly, we are inves-tigating the limitations of technologies. To be more con-structive, we believe that we can occasionally bring some new mathematical ideas into the business, which really can help to improve the identifi cation systems – and which to some extent also explore the possible limits of technologies.

Mi: As I said, some leading methods that are currently proposed have insuffi cient security. I explained why we are optimistic about the possibilities of improvement. It belongs to the purpose of research that one tries to prove lower bounds to the security that one can offer. This re-quires more work (in statistical modelling too). It is a dif-fi cult task that we take upon us.

But we cannot provide the certainty that a deployed system doesn’t have gaps in the security conception and guidelines. The conception of the system is very important and it has to reduce the risks in a hierarchical structure. I know from earlier experience in the industry that security is not only a matter of the cryptography used but also of the risk hierarchies. The fi rst project I developed was the security of online ATM systems in Switzerland. There you have a hierarchy of keys and the ultimate keys are, by de-sign, unknown to anybody. They are born, kept and used in a tamperproof security box. This is a computer whose memory is instantly deleted when you shake it or physi-cally tamper with it. The next hierarchy of keys are known to a very few well-trusted people, etc. Actually I believe that the management did not completely trust this tech-nology, so they also printed out the core key, making it ac-cessible to a designated bank director; maybe this is bet-ter. However, the most important keys should be known to a minimal number of people. This is a matter of design; it’s not a matter of cryptography or of biometrics.

There is a very strong connection between this research and the applied aspect, the aspect of concrete applica-tions in industry, in bank systems. You told us that the mathematical aspect is very important in biometrics. Are there concepts or aspects that you can take from pure mathematics and then apply in your research? Could you describe some of these?Mu: In our institute, we are of course not only concerned with biometrics. To give you an example, we have re-cently started to work on growth modelling of biological objects such as trees and leafs. This is supported by the German Science Foundation and we collaborate closely with colleagues from the forest science department, who perform fi eld experiments. This practical problem has initiated fundamental research towards principal com-ponent analysis on manifolds, which we call geodesic principal component analysis. Here we benefi t a lot from discussions with colleagues from differential geometry and optimization.

Mathematics and Applications

Pa: There’s one comment I’d like to make on this: the no-tions of pure mathematics and applied mathematics are not terribly fi rm or precise. When I was a student, applied mathematics was very much associated with physics but what is now stated as applied mathematics, for example in cryptography, is very, very much what in those days was considered as pure mathematics; it was number the-ory. Number theory was considered, let us say, in the late 1960s as one of the purest areas of mathematics; nowa-days it is one of the most applied areas of mathematics.

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Mi: Exactly, for instance the algebraic geometry over fi -nite fi elds that I mentioned.

Conversely, do you think that there is also some pay-back from applications of mathematics, fostering new theoretical research? Mi: Mathematics progresses both by the impulse from questions raised in physics – and nowadays from a much wider domain of applications – and from questions aris-ing in the mathematical research itself. An important as-pect is that the period it takes for some new fundamental mathematical insight to fi nd an application – a refl ection in the sensible world, I would say – is long, usually very much longer than, for instance, the time it takes for a discovery in experimental physics to be translated into a revolutionary technology.

Sometimes, this process is iterative. I am far from be-ing an expert but here I think for instance of Rieman-nian geometry used in relativity theory, whose further development has continuous impacts on mathematics and physics, as I understand it.

Pa: I think it actually seems to occur on a rather slow, almost geological, time scale. What happens is that you fi nd a student coming up, and she or he learns two or more different areas either from different teachers or from books or something like that and then brings them together in her or his person. And then the development takes place. Each generation of mathematicians has got this essentially dialectic aspect that brings together new connections and this is exactly what keeps mathematics alive. Just as a single line, it would die out. Public awareness – in Germany

2008 is the year of mathematics in Germany. How do you think the awareness of mathematics is in the real world, in the public?Mu: Personally, I would say that a major challenge for the mathematical community is to focus the attention of society on the benefi ts it draws from mathematical research. When something in mathematics is invented, typically it takes a very long time until it is recognized in society as a value or a contribution. Moreover, when a mathematical result is at the heart of a practical inven-tion it is not recognized as such anymore. In other dis-ciplines this goes faster and this process is much more direct. Inventions in molecular biology or in medicine, let’s say, are fi rst of all recognized as inventions of these disciplines. At the beginning of the 20th century Radon developed what nowadays is called the Radon transform, and about 60 or 70 years later people used it for tomog-raphy. Of course, nobody in the public related these two things anymore. And there are many other examples: the mathematics of fi nancial markets that is used in every investment bank nowadays is founded on the famous Ito-calculus from the early 50s, which itself is based on Brownian motion developed by Einstein and Wiener in the fi rst quarter of the last century. This has been very successfully applied in the 70s to option pricing, and the

Noble prize in economy was awarded for this develop-ment. These very long periods of time prevent the public from appreciating the value of mathematics and how it really matters to us.

Pa: Can I say something about my experience here? Much of modern mathematics started in Germany dur-ing the 19th century and it came to me as a huge surprise to discover how negative many people in Germany are about mathematics. It is a very strange experience: when one says one is a mathematician here, one usually gets the answer, ‘I was never any good at mathematics at school,’ or something of this nature. In fact, the attitude is quite different, let’s say, in the UK or in France where there are many programs in the media about mathematics, on the BBC or in other places. We’ve just been involved here in a program that is being made by Marcus du Sautoy for the BBC. One might hope, though it’s a rather weak hope, that in the course of the Year of Mathematics one might actually change this a little; it seems to be a specifi cally German attitude that you do not fi nd in other countries.

May I come to my last question – do you think that the special work you’re doing in biometrics, which com-bines some aspects of academics and some aspects of industry and applicability may have an effect on the recognition of mathematics in the public? Mi: If it succeeds in reaching the goals that we have set… Our goals are to improve technological systems, on the one hand, and to develop some theoretical models that can prove something positively in areas that are poorly understood, on the other hand. Maybe the best hope is that solutions found for this specifi c problem may call for some new mathematics, if we develop concepts that can be used in similar problems. This is the best I can hope for, and then let the waves go the way the waves go. It’s always a matter of who hears at the other end of the channel. It’s never the whole public.

Hanno Ehrler [[email protected]] studied musicol-ogy, art history and ethnology in Mainz. He works in part as a freelance journalist for the Frankfurter Allgemeine Zeitung (FAZ) and the public German TV station ARD (subjects: contemporary music, science fi ction, modern technologies).

Axel Munk [[email protected]] is a profes-sor at the Institut für Mathematische Stochastik at Göt-tingen University. His research in statistics focuses on non-parametric regression, medical statistics, statistical inverse problems and imaging, statistical modelling and shape analysis. Within pattern recognition, he is mainly interest-ed in the analysis of fi ngerprints.

Samuel J. Patterson [[email protected]] is a professor at the Mathematisches Institut, Göttingen Uni-versity. His main research areas comprise analysis on and around Kleinian groups, generalized theta functions and metaplectic groups, and analytic number theory in gen-eral.

Personal column


Personal columnPlease send information on mathematical awards and deaths to the editor.

Awards

The 2007 Oberwolfach Prize for Excellent Achievements in Algebra and Number Theory was awarded to Ngô Bao Châu (Paris-Sud/Orsay).

The 2008 Clay Research Awards has been given to Cliff Taubes (Harvard University, Cambridge MA, USA) for his proof of the Weinstein conjecture in dimension three, and to Claire Voi-sin (CNRS, IHES, and the Institut Mathématique de Jussieu, Paris, France) for her disproof of the Kodaira conjecture.

Gitta Kutyniok (Giessen University, Germany) has been select-ed to receive this year’s von Kaven Prize in Mathematics for her outstanding work in the fi eld of applied harmonic analysis. This prize is awarded by the von Kaven Foundation, which is administered by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).

Andreas Neuenkirch (University of Frankfurt, Germany) has been awarded the 2007 Information-Based Complexity Award for Young Researchers.

Newsletter editor Themistocles Rassias (National Technical University of Athens, Greece) has been awarded a Doctor Honoris Causa from the University of Alba Iulia (Romania). Congratulations!

László Lovász (Eötvös Loránd University, Budapest, Hungary and current president of the International Mathematical Union IMU) has been awarded the Bolyai Prize, one of the highest honours in Hungarian scientifi c life.

The 2008 Steele Prize for a Seminal Contribution to Mathe-matical Research has been awarded to Endre Szemerédi (Bu-dapest, Hungary, and Rutgers, USA) for the paper “On sets of integers containing no k elements in arithmetic progression”, Acta Arithmetica XXVII (1975).

One of the 2008 Maxime Bôcher Memorial Prizes has been awarded to Alberto Bressan (Italy and State College, PA, USA) for his fundamental works on hyperbolic conservation laws.

The 2008 Doob Prizes have been given to Enrico Bombieri (Italy and IAS Princeton, USA) and Walter Gubler (Switzerland and Humboldt University, Berlin, Germany) for their book “Heights in Diophantine Geometry” (Cambridge University Press, 2006).

The Autumn Prize of the Mathematical Society of Japan (MSJ) for 2007 has been awarded to Tadahisa Funaki (University of Tokyo, Japan) for his outstanding contribution to stochastic analysis on large scale interacting systems. The 2007 MSJ Geometry Prizes have been awarded to Shigeyuji Morita and Kenichi Yoshikawa (both from the University of Tokyo, Japan). The award to S. Morita has been made in rec-ognition of his fundamental research on mapping class groups;

the award to K. Yoshikawa has been made for his outstanding research on Ray-Singer analytic torsion and its behaviour on various moduli spaces.

The MSJ Analysis Prizes have been awarded to Shigeki Aida (Osaka University, Japan) for his contributions to stochastic analysis in infi nite dimensional spaces, to Toshiaki Hishida (Niigata University, Japan) for his contributions to new devel-opments in Fujita-Kato theory for the Navier-Stokes equations, and Takeshi Hirai (Kyoto University, Japan) for his contribu-tions to representation theory of infi nite symmetric groups.

Zbigniew Błocki (Kraków, Poland) was awarded the Zaremba Great Prize of the Polish Mathematical Society for his papers on complex analysis.

Grzegorz Świątek (Warsaw, Poland and Pennsylvania) was awarded the Banach Great Prize of the Polish Mathematical Society for his papers on dynamical systems.

Teresa Ledwina (Wrocław, Poland) was awarded the Steinhaus Great Prize of the Polish Mathematical Society for her papers on mathematical statistics.

Adam Skalski (Łódź, Poland) was awarded the Kuratowski Prize.

Radosław Adamczak (Warsaw, Poland) was awarded the Prize of the Polish Mathematical Society for young mathematicians.

Luis Barreira (Instituto Superior Técnico, Lisbon, Portugal) was the 2008 winner of the Ferran Sunyer i Balaguer Prize. He was acknowledged for his book “Dimension and Recurrence in Hyperbolic Dynamics”.

George Lusztig (Romania and MIT, Boston, USA) has been awarded the 2008 Steele Prize for Lifetime Achievement.

The Adams Prize of Cambridge University has been awarded jointly to Tom Bridgeland (University of Sheffi eld, UK) and David Tong (University of Cambridge, UK).

The 2007 SASTRA Ramanujan Prize has been awarded to Ben Green (University of Cambridge, UK). This annual prize, which was established in 2005, is for outstanding contributions to ar-eas of mathematics infl uenced by Srinivasa Ramanujan.

Ulrike Tillmann (Oxford University, UK) has been selected as the recipient of a Friedrich Wilhelm Bessel Research Award. This award is conferred in recognition of lifetime achievements in research.

Deaths

We regret to announce the deaths of:

Karl-Heinz Diener (Germany, 18 September 2007)Henry R. Dowson (UK, 28 January 2008)Anders Frankild (Denmark, 10 June 2007)Helmuth Gericke (Germany, 15 August 2007)Karl Gruenberg (UK, 10 October 2007)Samuel Karlin (UK, 18 December 2007)David Kendall (UK, 23 October 2007)Richard Lewis (UK, 26 July 2007)Farkas Miklos (Hungary, 28 August 2007)Georg Nöbeling (Germany, 16 February 2008)Alex Rosenberg (Germany and USA, 27 October 2007)Jean-Luc Verley (France, 25 October 2007)

Contents Volume 10 (2008)

Issue 1:J. Bourgain and W.-M. Wang, Quasi-periodic solutions of nonlinear random Schrödinger equationsEugenio Montefusco, Benedetta Pellacci and Marco Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systemsC. Bandle, J. v. Below and W. Reichel, Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions Marek Fila and Michael Winkler, Single-point blow-up on the boundary where the zero Dirichlet boundary condition is imposedItai Benjamini and Alain-Sol Sznitman, Giant component and vacant set for random walk on a discrete torusKoen Thas and Don Zagier, Finite projective planes, Fermat curves, and Gaussian periodsHoai-Minh Nguyen, Further characterizations of Sobolev spacesGabriele Mondello, A remark on the homotopical dimension of some moduli spaces of stable Riemann surfacesM. Farber and D. Schütz, Homological category weights and estimates for cat1(X, ξ)

Issue 2:Dong Li and Ya. G. Sinai, Blow ups of complex solutions of the 3D Navier–Stokes system and renormalization group methodUrsula Hamenstädt, Bounded cohomology and isometry groups of hyperbolic spacesMichael Larsen and Alexander Lubotzky, Representation growth of linear groupsI. P. Shestakov and E. Zelmanov, Some examples of nil Lie algebrasY.-P. Lee, Invariance of tautological equations I: conjectures and applicationsAndreas Cap, Infinitesimal automorphisms and deformations of parabolic geometriesJean Van Schaftingen and Michel Willem, Symmetry of solutions of semilinear elliptic problemsMargarida Mendes Lopes and Rita Pardini, Numerical Campedelli surfaces with fundamental group of order 9Sergiu Klainerman and Igor Rodnianski, Sharp L1 estimates for singular transport equationsM. Burak Erdogan, Michael Goldberg and Wilhelm Schlag, Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in �3

Leonid Makar-Limanov and Jie-Tai Yu, Degree estimate for subalgebras generated by two elementsJ.-P. Francoise, N. Roytvarf and Y. Yomdin, Analytic continuation and fixed points of the Poincaré mapping for a polynomial Abel equation

To appear:Bjorn Poonen,The moduli space of commutative algebras of finite rankJean Bourgain and Alex Gamburd, Expansion and random walks in SLd (�=pn�): IL. Escauriaza, C.E. Kenig, G. Ponce and L. Vega, Hardy’s uncertainty principle, convexity and Schrödinger EvolutionsTomek Szankowski, Three-space problems for the approximation propertyEitan Tadmor and Dongming Wei, On the global regularity of sub-critical Euler-Poisson equations with pressureIoan Bejenaru and Daniel Tataru, Large data local solutions for the derivative NLS equationRadu Ignat and Felix Otto, A compactness result in thin-film micromagnetics and the optimality of the Néel wallSergio Albeverio, Alexei Daletskii and Alexander Kalyuzhnyi, Random Witten Laplacians: Traces of Semigroups, L2-Betti Numbers and IndexJ. Krieger and W. Schlag, Non-generic blow-up solutions for the critical focusing NLS in 1-dJean Van Schaftingen, Estimates for L1-vector fields under higher-order differential conditions

Subscription informationJournal of the European Mathematical Society is published in one volume per annum, four issues per volume. Print ISSN: 1435-9855, online ISSN: 1435-9863The annual subscription rate is 320 Euro (suggested retail price, add 30 Euro for postage and handling). Other subscriptions on request.

Journal of the European Mathematical SocietyAims and Scope Journal of the European Mathematical Society (JEMS) is the official journal of the EMS. The Society, founded in 1990, works at promoting joint scien-tific efforts between the many different structures that characterize European mathematics. JEMS will publish research articles in all active areas of pureand applied mathematics. These will be selected by a distinguished, international board of editors for their outstanding quality and interest, accordingto the highest international standards. Occasionally, substantial survey papers on topics of exceptional interest will also be published. Starting in 1999,the Journal has been published by Springer-Verlag until the end of 2003. Since 2004 it is published by the EMS Publishing House. The first Editor-in-Chief of the Journal was J. Jost, succeeded by H. Brezis in 2003.

Editorial BoardHaïm Brezis (Editor-in-Chief), Université Pierre et Marie Curie, Paris, France,

Rutgers University, USA and Technion, Haifa, IsraelAntonio Ambrosetti, SISSA, Trieste, ItalyLuigi Ambrosio, Scuola Normale Superiore Pisa, ItalyEnrico Arbarello, Università di Roma “La Sapienza”, ItalyRobert John Aumann, The Hebrew University of Jerusalem, IsraelOle E. Barndorff-Nielsen, Aarhus University, DenmarkHenri Berestycki, EHESS, Paris, FranceJoseph Bernstein, Tel Aviv University, IsraelFabrice Bethuel, Université Pierre et Marie Curie, Paris, FranceJean Bourgain, Institute for Advanced Study, Princeton, USAJean-Michel Coron, Université Pierre et Marie Curie, Paris, FranceIldefonso Diaz, Instituto de España, Madrid, SpainCorrado De Concini, Università di Roma “La Sapienza”, ItalySimon Donaldson, Imperial College, London, UKYasha Eliashberg, Stanford University, USAGeoffrey Grimmett, Cambridge University, UKGerhard Huisken, Max-Planck-Institute for Gravitational Physics, Golm, Germany

Marius Iosifescu, Romanian Academy, Bucharest, RomaniaWilfrid S. Kendall, University of Warwick, Coventry, UKSergiu Klainerman, Princeton University, USAHendrik Lenstra, University of Leiden, The NetherlandsEduard Looijenga, Utrecht University, The NetherlandsAngus Macintyre, Queen Mary, University of London, UKIb Madsen, Aarhus University, DenmarkJean Mawhin, Université Catholique de Louvain, BelgiumStefan Müller, Max-Planck Institute for Mathematics in the Sciences, Leipzig, GermanySergey Novikov, University of Maryland, College Park, USA,

and Russian Academy of Sciences, Moscow, RussiaLambertus Peletier, University of Leiden, The NetherlandsAlfio Quarteroni, EPFL Lausanne, SwitzerlandMete Soner, Sabanci University, Istanbul, TurkeyAlain Sznitman, ETH Zürich, SwitzerlandMina Teicher, Bar-Ilan University, Ramat-Gan, IsraelClaire Voisin, IHES, Bures sur Yvette, FranceEfim Zelmanov, University of California, San Diego, USAGünter M. Ziegler, Technische Universität Berlin, Germany

European Mathematical Society Publishing House / Seminar for Applied Mathematics / ETH-Zentrum FLI C4CH-8092 Zürich, Switzerland / www.ems-ph.org / [email protected]

Interview


Interview with JEMS editor-in-chief Haïm Brezis Conducted by Thomas Hintermann (EMS Publishing House, Zurich, Switzerland)

What publishing experience did you have before you started working for JEMS?Shortly after I got my PhD, in the early seventies, I was invited to serve on the board of several journals, includ-ing the Journal of Functional Analysis, the Archive for Rational Mechanics and Analysis, and others. I learned much from the chief editors of those journals, particular-ly Jacques-Louis Lions, Paul Malliavin, Irving Segal and Jim Serrin. I am currently on the board of about 25 jour-nals, but of course with various levels of involvement. I initiated Communications in Contemporary Mathemat-ics ten years ago together with Xiao-Song Lin. Moreo-ver, I am a chief editor for two book series published by Birkhäuser and CRC Press.

How long have you been managing JEMS? I was asked in 2003 by Rolf Jeltsch, then President of the EMS, to take over from Jürgen Jost, the fi rst chief editor of JEMS, who had done a very good job but was eager to step down. This was around the time when the

EMS had decided to transfer JEMS from Springer to the newly created EMS Publishing House. At fi rst, I was re-luctant to accept, being already engaged with many other projects. I discussed the matter with some colleagues who were closely involved with the EMS, among them Mina Teicher, Doina Cioranescu and Jean-Pierre Bourguignon. They convinced me that it was important to carry on this project, which was originated by the fi rst president of the EMS, F. Hirzebruch. The change of publisher offered a unique opportunity to shape the journal and build on the good reputation the journal had already acquired before my time. Your own encouragement, Thomas, was another factor in my decision to accept the challenge.

Please describe your cooperation with the other editors and the refereeing process for JEMS.It is my principle to allow individual editors as much freedom in their activities as possible. In my experience, the more responsibility an editor of a journal is given, the more active and reliable he becomes. Usually, the editors of JEMS receive or solicit manuscripts and contact re-viewers entirely on their own, and then forward them to me with a well-documented recommendation. Of course, the ultimate decision lies in my hands but it is very rare that I challenge the advice of an editor. The editors are aware of this policy and they act with utmost care be-cause they feel responsible for the quality of JEMS. This is not always the policy for other journals; I know ex-amples where the chief editor has a much tighter hand on the process, often soliciting additional opinions that might confl ict with the original advice. As a result, mem-bers of the board become less involved and sometimes are even reluctant to communicate great papers out of fear that their enthusiasm might be challenged.

When I took over, I reshaped the board. My prima-ry concern was to bring in leading fi gures, at the peak of their creativity and connected to young people – the most natural source of papers! From my past experience with other journals I knew who was taking seriously his/her mission of editor; I avoided distinguished mathema-ticians who do not reply to email!

Since JEMS is a general mathematical journal, it was important to have a good distribution in terms of topics. Geography also played a role: one of our goals was to increase the visibility of the European Mathematical So-ciety within the international mathematical community. Most of the members of the board are based in Europe but we also have many European “expatriates” working in other countries, notably the United States. Everyone on the board understands that the excellent ranking of JEMS – among the top ten – is a direct result of his/her

Interview


personal activity. Moreover, my policy is to encourage our editors to publish some of their own work in JEMS. I know that this is considered “politically incorrect” in some editorial circles. But our journal is young and I feel that this is the best way to send a strong signal to authors about the quality of JEMS.

You are a prolifi c mathematical author (nonlinear functional analysis and PDEs and more) yourself. How do you fi nd the time necessary to manage the JEMS en-terprise?Finding the time is indeed a big problem and often re-quires compromise. On the other hand, I view this task as an integral part of my scientifi c activity. It also provides an excellent opportunity to become acquainted with new developments, particularly those outside my immediate fi eld of research. It is a unique observatory; the informa-tion I receive at JEMS becomes very useful when I take part in committees awarding prizes, e.g. at the Académie des Sciences.

Are journals still as important as in the past, in times where preprints are freely circulating on the Internet and in preprint archives? Please comment on the added value of refereeing!With all the preprints circulating via email, it is really hard to say where we are heading. Publishing in a respectable journal, and particularly in a top journal like JEMS, en-dows any paper with a quality stamp most authors fi nd very desirable. For young people it is often necessary on their CV. It is the ultimate sign that a paper has been found correct and worthy of attention by the mathemati-cal community. One should not underestimate the psy-chological impact on some talented people who work in isolation and sometimes under diffi cult conditions.

JEMS is a general mathematical journal, accepting papers from all areas. However, are there particularly strongly represented fi elds?My specialty is nonlinear PDEs and it is inevitable that my own fi eld is well represented. For example, we de-cided to publish a special issue dedicated to Antonio Ambrosetti, one of our board members, who is a leading fi gure in nonlinear PDEs. This tilted the balance towards analysis. However, we took care to readjust matters in later issues of the journal. I am closely acquainted with several editors from other fi elds and I encourage them to compensate my personal bias towards analysis. As it stands now, I think we can say that JEMS is a truly inter-disciplinary journal within mathematics.

Do you think that JEMS is representing the EMS in the eyes of mathematicians, or is it perceived as just a top journal disconnected from the society?It is the fate of most mathematical journals to be discon-nected from their “owners”. The journals of the AMS are not especially linked with the society in the eyes of the public. The same can be said of the CRAS or the Annales de l’ENS. Communications on Pure and Applied Math-ematics, which is an NYU journal, is an exception.

In the case of JEMS the situation is slightly different. The EMS is a rather young society closely connected with the emergence of Europe as a major scientifi c power and it is natural for the EMS to increase its visibility through a leading journal. I am pleased with the wide geographi-cal distributions of authors (about two thirds from Eu-rope and one third from the rest of the world). One of the goals of JEMS is to become the “fl agship” of the EMS.

In the last few years, JEMS has more than doubled the number of published papers, and the development is likely to continue. What makes JEMS so attractive for authors? What are, in your experience, the most impor-tant factors for authors when deciding on the journal for their article? The EMS is a not-for-profi t organiza-tion; do you think that this is an important detail for authors and editors?I suppose that the excellent reputation of the members of the board (as leading scientists) and their effi cient personalities plays a role. Also, many active young Euro-pean mathematicians have enjoyed the support of Euro-pean exchange networks in their formative years. They had the opportunity to meet and collaborate with col-leagues from other European countries. Not surprisingly, they have sympathy for a European journal. In addition, there is increasing discomfort among authors and editors toward the profi t-making giants in the publication indus-try; this is also a factor in favour of JEMS.

Any fi nal comments?I derive much pleasure from my activities at JEMS and the friendly collaboration of all the members of our board. The working relationship with the whole editorial team of the EMS has been superb, especially when a ma-jor overfl ow of high-class papers required a substantial increase in the size of the journal. I am very grateful to you, Thomas, for your continued enthusiasm during these “pioneering times” for JEMS.

Haïm Brezis [[email protected]] is professor emeritus at the Université Pierre et Marie Curie (Paris VI), visit-ing distinguished professor at Rutgers

University, USA, and at the Technion – Israel Institute of Technology. He is the editor-in-chief of the Journal of the European Mathematical Society.

Thomas Hintermann [[email protected]] is the director of the EMS Publishing House.

Mathematics Education


The 100th anniversary of ICMISymposium, Rome, 5–8 March 2008The fi rst century of the International Commission on Mathematical Instruction (1908–2008) Refl ecting and shaping the world of mathematics education

Maria G. (Mariolina) Bartolini Bussi (Modena, Italy)

At the beginning of March, in Rome, the centennial of the ICMI was celebrated. In this short paper, rather than covering all of the scientifi c components of the sympo-sium, I shall try to convey the emotion of being there. The website of the symposium (http://www.unige.ch/math/EnsMath/Rome2008/) contains a lot of informa-tion about the program and also photos of the scientifi c and the social events.

The symposium was held in two historical buildings: the Corsini Palace, which dates back to the 16th century and hosts the Academy of Lincei; and the Mattei Palace (again 16th

century), which hosts the Institute of Enciclopedia Italiana.The Academy of Lincei is the most ancient learned so-

ciety in the world. It was established as an international so-ciety in 1603 by Federico Cesi and others (a Dutch scientist among them). Galileo Galilei added his name and fame to the society a few years later (in 1611) and the number of academicians increased steadily with the addition of Ital-ians and non-Italians from the worlds of science, poetry, law and philology. In front of the Corsini Palace there is a villa called “The Farnesina”, built between the 15th and the 16th centuries; it has a beautiful garden and has famous Raffaello’s frescoes inside. The villa belongs to the acad-emy, which holds occasional formal celebrations there.

Both the Corsini and Mattei Palaces are in the city cen-tre, within walking distance of the Vatican and other famous sites in Rome. In particular, Corsini Palace is in Trastevere, a well-known district along the river Tevere and an area beloved by tourists; most of the district is for pedestrians only and visitors can enjoy the narrow streets with low, old houses and famous restaurants on the ground fl oor.

The ICMI was established in 1908, during the Interna-tional Congress of Mathematicians held in Rome, with the aim of supporting and expanding the interest of mathema-ticians in teaching in schools. Its fi rst president was Felix Klein. Something similar was attempted in many different subjects but only in mathematics was there success in ob-taining widespread international collaboration, in order to face problems relating to the social image of mathe-matics, to diffi culties in learning and to links with research and applications. Some years ago the idea of celebrating

the centennial in the same place where the commission had been established was launched. In spite of the many diffi culties of hosting a large congress in a city crowded with tourists all through the year, Ferdinando Arzarello and Marta Menghini accepted the challenge and designed a celebration that aimed to evoke the original event as much as possible: only a few days separated the true birth-day from the dates of the symposium; the palace was the same (the Palace of the Academy of Lincei); and even the social program was the same, with a beautiful banquet and excursion to the famous villas of Tivoli (Villa Adriana and Villa d’Este). The scientifi c committee (chaired by Arza-rello) was composed of researchers from every continent who were well-known fi gures in the fi eld of the didactics of mathematics, both for the research that they had car-ried out and for the institutional positions they held. The local organising committee was made up of professors from Italian Departments of Mathematics.

Everything worked in a wonderful way. It was not easy in Rome to leave the beautiful surroundings to go into the Corsini Palace to take part in the symposium. Yet the pro-gram was so interesting that the large room of the meeting was always crowded.

When a society turns one hundred, the memories are usually to be reconstructed by historians. Yet the organizers succeeded in interviewing (and in most cases also inviting to Rome) some of the most relevant mathematicians and mathematics educators who bear witness to this long his-tory. The culmination of effort for the event has produced a website that will constitute an extraordinary source for the future (http://www.icmihistory.unito.it/). In the section ‘Interviews and fi lm clips’ many eyewitnesses utter their thoughts, e.g. Emma Castelnuovo, Trevor Fletcher, Mau-rice Glaymann, Geoffrey Howson, Jean-Pierre Kahane, Heinz Kunle, André Revuz and Bryan Thwaites. Others have been evoked by the invited lecturers. Hyman Bass (the ex-president of the ICMI) opened the symposium with a speech on ‘Moments in the history of the ICMI’. The closing plenary was given by Michèle Artigue, the president of the ICMI, on the theme ‘One century at the interface between mathematics and mathematics education – refl ections and perspectives’. Between them the follow-ing themes were addressed in plenary speechs: The devel-opment of mathematics education as an academic fi eld; In-tuition and rigour in mathematics education; Perspectives on the balance between application & modelling and ‘pure’ mathematics in the teaching and learning of mathematics; The relationship between research and practice in mathe-matics education – international examples of good practice; The origins and early incarnations of the ICMI; the ICMI Renaissance – the emergence of new issues in mathemat-ics education; and Centres and peripheries in mathematics education, the ICMI’s challenges and future.

This list of titles conveys the idea that the celebration was not just reminiscing in history but was open to the future directions of research in mathematics education and the possible action to be taken to improve the level of scientifi c culture in various countries. Other scientifi c activities were the working groups and short talks. Details may be found on the website mentioned above.

The logo of the symposium reproduces the fl oor of the Capitol Square in Rome, which was designed by Michelangelo.

Mathematics Education


More than 180 invited participants were present in Rome from all over the world. Besides the representatives of the Italian institutions that supported the event and some of the past offi cers of the ICMI, it is worthwhile to note the attendance of the president Lazlo Lovasz and the vice-president Claudio Procesi of the IMU and of the president of the ICTP (International Centre for Theoretical Physics) Ramadas Ramakrishnan, on behalf of UNESCO. Most participants were accompanied by relatives and friends, as Rome is always appealing for a spring holiday.

The proceedings are in progress and will be ready in a few months. There is no doubt that they will constitute an indefeasible source for all the mathematics educators who feel the need to know about the roots of their academic fi eld and also for the organizers of the next centennial symposium in Rome in 2108!

ICMI AWARDSIn 2000, the ICMI decided to create two awards for re-search in mathematics education: the Felix Klein Medal, from the name of the fi rst president of the ICMI, and the Hans Freudenthal Medal, from the name of its eighth pres-ident. The fi rst two medals were announced in 2003 and were awarded to Guy Brousseau, France (Klein medal) and Celia Hoyles, UK (Freudenthal medal). The citations are available at http://www.icme10.dk/pages/081icmi-award.htm. The medallists for 2005 were Ubiratan D’Ambrosio, Brazil (Klein medal) and Paul Cobb, USA (Freudenthal medal). The citations are available at http://www.univ-par-is-diderot.fr/2006/04-icmi.pdf. The medallists for 2007 are Jeremy Kilpatrick, USA (Klein medal) and Anna Sfard, Israel (Freudenthal medal). At ICME11 (Monterrey, Mex-ico, 5-13 July 2008) the last four medals (2005-2007) will be presented. Short citations of the 2007 medals follow:

Jeremy Kilpatrick, University of Georgia, Athens, GA, USAIt is with great pleasure that the ICMI Awards Committee hereby announces that the Felix Klein Medal for 2007 is given to Professor Jeremy Kilpatrick, Uni-versity of Georgia, Athens, GA, USA, in

recognition of his more than forty years of sustained and distinguished lifetime achievement in mathematics educa-tion research and development. Jeremy Kilpatrick’s nu-merous contributions and services to mathematics educa-tion as a fi eld of theory and practice, as he prefers to call it, are centred around his extraordinary ability to refl ect on, critically analyse and unify essential aspects of our fi eld as it has developed since the early 20th century, while always insisting on the need for reconciliation and balance among the points of view taken, the approaches undertaken and the methodologies adopted for research. It is a character-istic feature of Jeremy Kilpatrick that he has always em-braced a very cosmopolitan perspective on mathematics education. Thus he has worked in Brazil, Colombia, El Sal-vador, Italy, New Zealand, Singapore, South Africa, Spain, Sweden and Thailand, in addition to being, of course, ex-traordinarily knowledgeable about the international liter-ature. Throughout his academic career, Jeremy Kilpatrick

has published groundbreaking papers, book chapters and books – many of which are now standard references in the literature – on problem solving, on the history of research in mathematics education, on teachers’ profi ciency, on cur-riculum change and its history, and on assessment.

Anna Sfard, University of Haifa, Israel, and the Institute of Education, University of London, UK (also affi liated to Michigan State University).It is with great pleasure that the ICMI Awards Committee hereby announces that the Hans Freudenthal Medal for 2007

is given to Professor Anna Sfard, University of Haifa, Is-rael, and the University of London, UK, in recognition of her highly signifi cant and scientifi cally deep accomplish-ments within a consistent, long-term research programme focused on objectifi cation and discourse in mathematics education, which has had a major impact on many strands of research in mathematics education and on numerous young researchers. In addition to publications related to the above-mentioned research programme, Anna Sfard has published numerous other papers and book chapters within a broad range of topics. It is a characteristic fea-ture of Anna Sfard’s scientifi c achievements that they are always very thorough, original and intellectually sharp. She often uncovers the tacit if not hidden assumptions behind notions, approaches and conventional wisdom, and by turning things upside-down she usually succeeds in generating new fundamental and striking insights into complex issues and problems.

Mogens Niss, Chair, The ICMI Awards Committee, [email protected].

People who are interested in being informed about ICMI activities are invited to subscribe to the ICMI News (free). There are two ways of subscribing to ICMI News:1. Click on http://www.mathunion.org/ICMI/Mailinglist with a Web browser and go to the “Subscribe” button to subscribe to ICMI News online.2. Send an email to [email protected] with the subject-line: Subject: subscribe.Previous issues can be seen at http://www.mathunion.org/pipermail/icmi-news.

In the issue 67 of this Newsletter (p. 18) the ICMI study n. 19 on Proof and Proving in Mathematics Education was announced. The address of the website of the study has been changed and is now: http://www.icmi19.comThe deadline for submissions is still June 30, 2008.

Mariolina Bartolini Bussi [[email protected]] is the Newsletter Editor within Mathematics Education. A short biogra-phy can be found in issue 55, page 4.

ERCOM


The Mathematisches Forschungsinstitut Oberwolfach (MFO) was founded in 1944 and has developed over the years into an international research centre. In mathemati-cal research, interchange of ideas plays a central role. The high degree of abstraction of mathematics and the com-pact way it is presented necessitate direct personal com-munication. Although most new results are nowadays quickly made available to the mathematical community via electronic media, this cannot replace personal contact among scientists. The importance of personal contact in-creases with the constant increase of specialization.

Therefore, the institute concentrates on cooperative research activities of larger (workshop programme) or smaller (mini-workshop programme and Research in Pairs) groups. Leading representatives of particularly relevant research areas from all over the world are invit-ed to Oberwolfach (about 30% coming from Germany and 40% from elsewhere in Europe) offering them the opportunity to pursue their research activities. The insti-tute’s premises offer free accommodation (full board) for about 55–60 invited visitors. In all activities, participation of promising young scientists plays an important role.

Scientifi c ProgrammesThe MFO has two large, central tasks: the weekly work-shop programme and the Research in Pairs programme for longer-term research stays. Additionally, there are fur-ther activities of the MFO that address young research-ers. The scientifi c programme is operated annually over 50 weeks and covers all areas in mathematics, including applications.

1. The Workshop Programme. The main scientifi c pro-gramme consists of about 40 week-long workshops per year, each with about 50 participants. Alternatively, there

ERCOM: Mathematisches Forschungsinstitut Oberwolfach

can be two parallel workshops of half that size (about 25 participants). The workshops are organised by inter-nationally leading experts of the respective fi elds. The participants are personally invited by the director after recommendations by the organizers. A special character-istic feature of the Oberwolfach Workshops is a research orientation. Very often the guest researchers appreciate the stimulating atmosphere. Many signifi cant research projects owe their origins to the realisation of a workshop in Oberwolfach.

2. The Mini-Workshop Programme. This programme offers 12 week-long mini-workshops per year, each with about 15 participants. These mini-workshops are aimed especial-ly at junior researchers who have a focused subject, and allow them to react on recent developments since the sub-jects are fi xed only half a year before the mini-workshop takes place. 3. The Oberwolfach Arbeitsgemeinschaft. The idea of the Arbeitsgemeinschaft for young as well as for senior re-searchers is to learn about a new active topic by giving a lecture, guided by leading international specialists. The Arbeitsgemeinschaft meets twice a year for one week and is organized by Professor Christopher Deninger and Pro-fessor Gerd Faltings.

4. The Oberwolfach Seminars. The Oberwolfach Seminars are week-long events taking place six times a year. They are organised by leading experts in the fi eld and address post-docs and PhD students from all over the world. The aim is to introduce 25 participants to a particularly hot development.

5. The Research in Pairs Programme. The Research in Pairs (RiP) Programme aims at small groups of 2–4 researchers from different places working together at the Mathema-tisches Forschungsinstitut Oberwolfach for anything from 2 weeks to 3 months on a specifi c project.

6. Oberwolfach Leibniz Fellows. The focus of this new postdoctoral programme, which has been set up for the fi rst time from 2007 to 2010, is to support excellent young researchers in an important period of their scientifi c ca-reer by providing ideal working conditions in an interna-tional atmosphere. Outstanding young researchers can ap-ply to carry out a research project, individually or in small groups, for a period from two to six months. They may propose for co-workers to visit Oberwolfach for shorter periods. Oberwolfach Leibniz Fellows should be involved in an active research group at a university or another re-search institute. It is also possible to make an application to the European Post-Doctoral Institute (EPDI).

Henri Cartan at the original Oberwolfach building

D E G R U Y T E R

*For orders placed in North America. Prices are subject to change.

Prices do not include postage and handling.

Victor Zvyagin / Dmitry Vorotnikov

■ Topological Approximation Methods for Evolutionary Problems of Nonlinear HydrodynamicsJune 2008. XII, 230 pages. Hardcover.

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The authors present functional analytical methods for solving a class of partial differential equations. The re-

sults have important applications to the numerical treatment of rheology (specific examples are the behaviour

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Barbara Kaltenbacher / Andreas Neubauer / Otmar Scherzer

■ Iterative Regularization Methods for Nonlinear Ill-Posed ProblemsMay 2008. Approx. VIII, 196 pages. Hardcover.

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Radon Series on Computational and Applied Mathematics 6

Nonlinear inverse problems appear in many applications, and typically they lead to mathematical models that

are ill-posed, i.e., they are unstable under data perturbations. Those problems require a regularization, i.e., a

special numerical treatment. This book presents regularization schemes which are based on iteration methods,

e.g., nonlinear Landweber iteration, level set methods, multilevel methods and Newton type methods.

Ulrich Kulisch

■ Computer Arithmetic and ValidityTheory, Implementation, and ApplicationsMay 2008. Approx. XVIII, 410 pages. Hardcover.

RRP € [D] 78.00 / * US$ 108.00.

ISBN 978-3-11-020318-9

de Gruyter Studies in Mathematics 33

The present book deals with the theory of computer arithmetic, its implementation on digital computers and

applications in applied mathematics to compute highly accurate and mathematically verified results. The aim

is to improve the accuracy of numerical computing (by implementing advanced computer arithmetic) and to

control the quality of the computed results (validity). The book can be useful as high-level undergraduate text-

book but also as reference work for scientists researching computer arithmetic and applied mathematics.

Hans-Otto Georgii■ StochasticsIntroduction to Probability and StatisticsTransl. by Marcel Ortgiese / Ellen Baake / Hans-Otto GeorgiiFebruary 2008. IX, 370 pages. Paperback.

RRP € [D] 39.95 / * US$ 49.00.

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de Gruyter Textbook

This book is a translation of the third edition of the well accepted German textbook ‘Stochastik’, which presents

the fundamental ideas and results of both probability theory and statistics, and comprises the material of a

one-year course. The stochastic concepts, models and methods are motivated by examples and problems and

then developed and analysed systematically.

www.degruyter.com

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Publications and Further Activities

1. The Oberwolfach Reports (OWR) began in 2004 as a new series of publications of the institute in collaboration with the EMS Publishing House. The four issues comprise more than 3000 pages per year. The OWR are formed of the offi cial reports of every workshop, containing extend-ed abstracts of the given talks, from one to three pages per talk, including references. The aim is to report periodically upon the state of mathematical research, and to make these reports available to the mathematical community.

2. The book series Oberwolfach Seminars (OWS) is pub-lished in cooperation with Birkhäuser. In this series, the material of the Oberwolfach seminars is made available to an even larger audience.

3. The Oberwolfach Preprints (OWP) mainly contain re-search results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs Programme and the Oberwolfach Leibniz Fellows.

4. Once every three years the Oberwolfach Prize is award-ed by the Gesellschaft für mathematische Forschung e.V. and by the Oberwolfach Stiftung to young mathemati-cians. The prize is awarded for excellent achievements in changing fi elds of mathematics. 5. The John Todd Fellowship is awarded every three years by the Oberwolfach Foundation and the Mathematisches Forschungsinstitut Oberwolfach to young excellent math-ematicians working in numerical analysis.

6. On a two year rotation, a training week for school teach-ers (and librarians on the alternate years) of the State of Baden-Württemberg takes place.

7. The institute also hosts the annual fi nal training week for especially gifted German pupils to prepare for the In-ternational Mathematical Olympiad.

Buildings

The buildings, which were mainly fi nanced by the Volkswagen-Stiftung, do not only offer accommodation facilities for visitors but also form an excellent and stimu-lating frame for research activities at the institute. Be-sides the well-adapted scientifi c infrastructure it is also the institute’s remote location, and the excellent service with board and lodging in the guest house close to the conference and library building, that guarantees effi cient and concentrated working conditions for the guests.

The library building is located immediately downhill from the guest house. It has replaced the former build-ing (the old castle) and was inaugurated in 1975. Com-prising the library, two lecture halls, several small dis-cussion rooms and numerous computer stations, it offers excellent working conditions for scientifi c research. The offi ces of the scientifi c administration are also part of

this building. The library, the discussion rooms and the cafeteria are open day and night and offer scientists the possibility to work at any time, an option that is eagerly taken up. The guest house was inaugurated in 1967. Each scientist is housed in a single room with its own bathroom. In addi-tion, eight apartments and fi ve bungalows enable a longer stay at the MFO within the Research in Pairs and the Oberwolfach Leibniz Fellows programmes. Due to their age, the maintenance of the buildings is of greatest impor-tance. The refurbishment of the guest house and the bun-galows is going on until 2010.

Infrastructure

The basis for the successful realisation of the research programmes described above is an excellent infrastruc-ture. Traditionally in mathematics, the library plays the main role. The Oberwolfach Library, which some years ago was ranked by the AMS as the best mathematical library outside the USA, contains about 45,000 issues of books, proceedings, etc. and subscribes to approximately 500 print journals and 3,000 electronic journals. The li-brary can be used by the guest researchers 24 hours a day. By an enlargement in 2007, which was fi nanced by the Klaus Tschira Foundation and the Volkswagen Founda-tion, the MFO is able to provide new capacity for about the next 20 years.

During the last few years the computer pool has been improved signifi cantly and is now up-to-date.

The MFO owns a large collection of more than 9,000 photographs of mathematicians, which are also avail-able online. Professor Konrad Jacobs, Erlangen, is one of those who have contributed to this photo collection. Spe-cial thanks go to Springer-Verlag Heidelberg for helping organize the digitalization of the photos.

During the last twenty years, mathematical software has developed into an established tool for mathematical research and education. In some fi elds, its importance is comparable to that of mathematical literature. However, collections of information about mathematical software so far only exist in a rudimentary form. The intention of the Oberwolfach References on Mathematical Software (ORMS) project is to fi ll this gap. This includes a web-in-terfaced collection of detailed information and links and a classifi cation scheme for mathematical software even-tually aiming to cover all thematic aspects of mathemati-cal software.

Institutional Structure and Statutes of the MFO

Since 2005, the MFO has been registered as a non-profi t corporation (gemeinnützige GmbH). The MFO is headed by a director supported by a vice-director. The sole associ-ate of the MFO is the Gesellschaft für Mathematische For-schung e.V. (GMF), represented by its board. The GMF is also a non-profi t society and was founded in Freiburg, 1959, in order to set up legal representation and scientifi c back-

Book review


ing for the institute. Financing of the MFO is shared by the Federal Republic of Germany and the Federal States with emphasis on the local state of Baden-Württemberg. Being a member of the Leibniz-Gemeinschaft is a prerequisite for the common fi nancing. The fi nancial partners are rep-resented in the Administrative Council of the MFO, which in its function as most important supervisory panel decides on the medium-term and long-term fi nance and budget planning. The institute and the administrative council are supported by the Scientifi c Advisory Board, which is com-posed of 6-8 internationally renowned mathematicians.

The statutes of the MFO and of the GMF contain the following aims, to be achieved with an international scope:

- The promotion of research in mathematics.- The intensifi cation of scientifi c collaboration. - The intensifi cation of education and training in math-

ematics and related areas.- The promotion of young scientists.

The director of the institute decides on the scientifi c pro-gramme in cooperation with the scientifi c committee of the GMF. For the scientifi c programme, this is the most important panel of the institute. It is based on the honor-ary work of about 20 mathematicians, covering all areas of mathematics. The scientifi c committee examines all scientifi c events at the institute prior to their approval. The programme is fi xed in a competitive procedure ac-cording to strictly scientifi c criteria.

For the planning and realization of the scientifi c pro-gramme of the MFO approximately 20 staff positions are

provided in various divisions, such as scientifi c and ad-ministration management, library, IT-service, guest serv-ice and housekeeping.

The Förderverein (Friends of Oberwolfach) of the MFO has more than 700 members and provides addi-tional fi nancial support for the MFO by its membership fees. The Oberwolfach Foundation, a foundation of pub-lic utility within the Förderverein, provides long-term fi nancial support by building up an endowment. Within the Oberwolfach Stiftung the Horst Tietz Fund plays an important role by providing special funds.

Please consult the web side of the MFO (www.mfo.de) for further, more detailed information concerning the scientifi c programme and the various committees and boards. A short guide for applications is online at http://www.mfo.de/programme/ProposalGuide2008.pdf .

Book reviewArne Jensen (Aalborg, Denmark)

Att våga sitt tärningskastGösta Mittag-Leffl er 1846–1927Arild StubhaugTranslated from Norwegian by Kjell-Ove WidmanAtlantis 2007ISBN 978-91-7353-185-6

This book is a biography of Gösta Mittag-Leffl er com-prising more than 750 pages. It was written by Arild Stubhaug, who is also the author of biographies of Niels Henrik Abel and Sophus Lie and originally pub-lished in Norwegian under the title “Med viten og vilje”

(Aschehoug 2007); an English edition is in preparation with Springer-Verlag and due in 2009. Arild Stubhaug has a background both in mathematics and literature, making him ideally suited to embark on the monumen-tal task of writing a biography of such a complex and at times controversial person as Gösta Mittag-Leffl er.

The book starts in the middle of events, with a de-scription of a visit to Egypt by Gösta Mittag-Leffl er and his wife Signe. They were also accompanied by a personal physician. Mittag-Leffl er was 53 years old and his wife was 38. The trip lasted from the end of 1899 until Easter 1900. The main reason for the trip was Mittag-Leffl er’s health problems (in particular stom-ach problems). This is a recurring theme throughout the biography. The dry desert air seems to have had a benefi cial infl uence.

As can be inferred from this short summary of the introductory chapter, the biography brings us very close to Mittag-Leffl er and his daily life, routines and cares. The biography is based on the wealth of material that Mittag-Leffl er left behind. Some documents are kept in the National Library of Sweden (Kungliga biblioteket), Stockholm, while other documents are at the Mittag-Leffl er Institute in the suburb of Djursholm. The docu-

Participants of the 2004 mini workshop Geometry and Duality in String Theory

Book review


ments include diaries, letters, newspaper clippings, etc. Arild Stubhaug has carefully researched this wealth of material and distilled out of it a fascinating account of a complex personality.

Mittag-Leffl er was born in 1846, the son of Johan Olof Leffl er and Gustava Wilhelmina, née Mittag. He had one sister Anne Charlotte (born 1849) and two brothers Fritz (born 1847) and Arthur (born 1854). His father devel-oped a mental disorder and was committed to care from 1870 until his death. This illness might be one of the rea-sons why Gösta Mittag-Leffl er had no children. He was very close to his mother and wrote her many letters. From these letters (and the replies) Arild Stubhaug has gained access to some of his personal thoughts during formative periods of his life. This has given a view of him that would otherwise not be available.

Mittag-Leffl er received his doctorate from Uppsala University in 1872. His thesis was on some results in complex analysis. One characteristic of this biography is that it does not provide any details on his mathematical achievements. As a mathematician one may regret this. However, it is not for his mathematical achievements that he deserves a well researched biography.

Probably one of the most important events in his life was the award of a travel grant (Bysantinska resestipend-iet) in 1873, allowing him to visit mathematicians in Paris and Berlin. In Paris he visited Hermite and also became acquainted with other French mathematicians. These contacts were to play an important role in his career, for example leading to a close acquaintance with Hermite’s student Poincaré. After a semester in Paris he moved to Berlin and met Weierstrass. This connection was to deter-mine the direction of many of his activities in the future. In particular, he heard about Sonja Kovalevsky for the fi rst time.

Mittag-Leffl er was called to a professorship at the newly established Stockholms Högskola (later Stock-holm University) in 1881 and from that time his base was Stockholm. However he kept up contacts with a large number of European mathematicians during the rest of his career. He often travelled and reports on his travels take up a large part of the biography. See the accompa-nying translation of a selected chapter in this newsletter issue.

Many of the high points of his career after returning to Stockholm are probably known to many mathemati-cians. He managed to get Sonja Kovalevsky appointed to a professorship at Stockholms Högskola, he founded the journal Acta Mathematica, and he educated and promoted a number of brilliant Swedish mathemati-cians, including I. Fredholm. All this, and much more, is meticulously related. Many details are added that most of us probably never knew. For example that the econo-my of Acta Mathematica was precarious for many years and that Mittag-Leffl er paid part of the expenses from his own funds.

Apart from mathematics Mittag-Leffl er also had a career as a business man. He invested in many busi-nesses and made and lost money. This is a fascinating part of his life that is exposed in detail for the fi rst time. Again the wealth of written material makes this pos-sible.

At some points in his career he was very wealthy. To-wards the end of his life he had very little left. In 1916 on his 70th birthday he and his wife Signe announced in a testament the formation of Makarna Mittag-Leffl ers Matematiska Stiftelse under the Royal Academy. The purpose was the formation of a mathematical research institute.1 The donation included his library and his villa in Djursholm. Extracts of the testament were published in Acta Mathematica.

Downturns in his fi nancial situation and other events meant that the vision was not realized for many years. It is fortunate that the donation in 1916 of the library and the villa put those out of reach of his creditors. It was only in the late 1960s that Lennart Carleson managed to secure funding for turning the villa in Djursholm into a world class mathematical research institute.

The impression one gets after reading this compre-hensive account is of a very complex personality. So many facets are revealed that it is clear he must have been at times both very charming and able to get close to peo-ple of very different personalities, for example Hermite, Painlevé and Weierstrass, and at other times not a person one would like to go up against.

I will not say anything about Gösta Mittag-Leffl er and Alfred Nobel. To fi nd out about this you will have to read the book. The account given changed the picture I had of the two and their relations.

The reader is also rewarded with a lively picture of the life and social activities of the upper segment of Swedish society during that period. It is important to see the mathematics and the mathematicians in their con-temporary context.

I enjoyed reading this book very much and hope you will do the same.

Arne Jensen [[email protected]] got his PhD from the University of Aarhus in 1979. He has been a pro-fessor of mathematics at Aalborg University since 1988. He served as acting director of the Mittag-Leffl er Insti-tute from 1993 to the beginning of 1995. In 2000–01 he was a visiting professor at the University of Tokyo. His research interests are spectral and scattering theory for Schrödinger operators.

1 A report on the present Mittag-Leffl er Institute appeared in the Newsletter issue 56 (June 2005), pp. 29–30.

Conferences


Forthcoming conferencescompiled by Mădălina Păcurar (Cluj-Napoca, Romania)

Please e-mail announcements of European conferences, work-shops and mathematical meetings of interest to EMS members, to one of the addresses [email protected] or ma-dalina_ [email protected]. Announcements should be written in a style similar to those here, and sent as Microsoft Word fi les or as text fi les (but not as TeX input fi les).

June 2008

1–7: Applications of Ultrafi lters and Ultraproducts in Mathematics (ULTRAMATH 2008), Pisa, ItalyInformation: [email protected];http://www.dm.unipi.it/~ultramath

2–6: International Conference on Random Matrices (ICRAM), Sousse, TunisiaInformation: [email protected];http://www.tunss.net/accueil.php?id=ICRAM

2–6: Thompson’s groups: new developments and inter-faces, CIRM Luminy, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

3–6: Chaotic Modeling and Simulation International Con-ference (CHAOS2008), Chania, Crete, GreeceInformation: [email protected]; http://www.asmda.net/cha-os2008

5–6: 12th Galway Topology Colloquium, Galway, IrelandInformation: [email protected];http://www.maths.nuigalway.ie/conferences/topology08.html

6–11: Tenth International Conference on Geometry, Inte-grability and Quantization, Sts. Constantine and Elena resort, Varna, BulgariaInformation: [email protected]; http://www.bio21.bas.bg/conference/

8–14: Mathematical Inequalities and Applications 2008, Trogir – Split, CroatiaInformation: [email protected]; http://mia2008.ele-math.com/

8–14: 34th International Conference “Applications of Math-ematics in Engineering and Economics”, Resort of Sozo-pol, BulgariaInformation: mtod@tu-sofi a.bg;http://www.tu-sofi a.bg/fpmi/amee/index.html

9–11: International Conference “Modelling and Compu-tation on Complex Networks and Related Topics” (Net-Works 2008), Pamplona, SpainInformation: [email protected]; http://www.fi sica.unav.es/networks2008/default.html

9–13: Geometric Applications of Microlocal Analysis, CIRM Luminy, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

9–13: Conference on Algebraic and Geometric Topology, Gdańsk, PolandInformation: [email protected]; http://math.univ.gda.pl/cagt/

9–13: Educational Week on Noncommutative Integration, Leiden, NetherlandsInformation: [email protected]; http://www.math.leidenuniv.nl/~mdejeu/NoncomIntWeek_2008

9–14: II International Summer School on Geometry, Me-chanics and Control, La Palma (Canary Islands), SpainInformation: [email protected]; http://webpages.ull.es/users/gmcnet/Summer-School08/home1.htm

9–14: Colloquium of Non Commutative Algebra, Sher-brooke, Québec, CanadaInformation: [email protected]; http://www.crm.umontreal.ca/~paradis/Sherbrooke/index_e. shtml

9–19: Advances in Set-Theoretic Topology: Conference in Honour of Tsugunori Nogura on his 60th Birthday, Erice, Sicily, Italy Information: [email protected];http://www.math.sci.ehime-u.ac.jp/erice/

9–20: GTEM/TUBITAK Summer School: Geometry and Arithmetic of Moduli Spaces of Coverings, Istanbul, Turkey Information: [email protected];http://math.gsu.edu.tr/GAMSC/index.htm

15–22: ESF-MSHE-PAN Conference on Operator Theory, Analysis and Mathematical Physics, Będlewo, PolandInformation: [email protected]; http://www.esf.org/conferences/08279

15–24: CIMPA-School Nonlinear analysis and Geometric PDE, Tsaghkadzor, ArmeniaInformation: [email protected]; http://www.imprs-mis.mpg.de/schools.html

16–17: Fifth European PKI Workshop, Trondheim, NorwayInformation: [email protected];http://www.item.ntnu.no/europki08/

16–19: 2nd International Conference on Mathematics & Statistics, Athens, GreeceInformation: [email protected];http://www.atiner.gr/docs/Mathematics.htm 16–20: Workshop on population dynamics and mathemat-ical biology, CIRM Luminy, Marseille, France Information: [email protected]; http://www.cirm.univ-mrs.fr

16–20: Homotopical Group Theory and Homological Alge-braic Geometry Workshop, Copenhagen, DenmarkInformation: http://www.math.ku.dk/~jg/homotopical2008/

Conferences


16–20: Fourth Conference on Numerical Analysis and Ap-plications, Lozenetz, BulgariaInformation: http://www.ru.acad.bg/naa08/

16–20: Conference on vector bundles in honour of S. Ra-manan (on the occasion of his 70th birthday), Mirafl ores de la Sierra (Madrid), SpainInformation: [email protected];http://www.mat.csic.es/webpages/moduli2008/ramanan/

16–20: 15-th Conference of the International Linear Alge-bra Society (ILAS 2008), Cancún, MexicoInformation: [email protected]; http://star.izt.uam.mx/ILAS08/

16–20: The 11th Rhine Workshop on Computer Algebra, Levico Terme, Trento, ItalyInformation: [email protected];http://science.unitn.it/~degraaf/rwca.html

16–21: International Scientifi c Conference „Differential Equations, Theory of Functions and their Applications“ dedicated to 70th birthday of academician of NAS of Ukraine A.M.Samoilenko, Melitopol, UkraineInformation: [email protected];http://www.imath.kiev.ua/conf2008/

17–20: Structural Dynamical Systems: Computational As-pects, Capitolo-Monopoli, Bari, Italy Information: [email protected];http://www.dm.uniba.it/~delbuono/sds2008.htm

17–22: International conference “Differential Equations and Topology” dedicated to the Centennial Anniversary of Lev Semenovich Pontryagin, Moscow, RussiaInformation: [email protected]; http://pont2008.cs.msu.ru

22–27: CR Geometry and PDE’s – III, Levico Terme, Trento, ItalyInformation: [email protected];http://www.science.unitn.it/cirm/AnnCR2008.html

22–28: Combinatorics 2008, Costermano (VR), Italy Information: [email protected]; http://combinatorics.ing.unibs.it

23–27: Homotopical Group Theory and Topological Alge-braic Geometry, Max Planck Institute for Mathematics Bonn, Germany Information: [email protected];http://www.ruhr-uni-bochum.de/topologie/conf08/

23–27: Hermitian symmetric spaces, Jordan algebras and related problems, CIRM Luminy, Marseille, France Information: [email protected]; http://www.cirm.univ-mrs.fr

23–27: Conference on Differential and Difference Equa-tions and Applications 2008 (CDDEA 2008), Strečno, Slovak Republic Information: [email protected]; http://www.fpv.uniza.sk/cddea/

23–27: Workshop on Geometric Analysis, Elasticity and PDEs, on the 60th Birthday of John Ball, Heriot Watt Univer-sity, Edinburgh, UKInformation: [email protected];http://www.icms.org.uk/workshops/pde

23–27: First Iberoamerican Meeting on Geometry, Me-chanics, and Control, Santiago de Compostela, SpainInformation: [email protected];http://www.gmcnetwork.org/eia08

23–27: Phenomena in High Dimensions, 4th Annual Con-ference, Seville, SpainInformation: [email protected];http://www.congreso.us.es/phd/

24–26: Current Geometry, the IX Edition of the Interna-tional Conference on problems and trends of contempo-rary geometry, Naples, ItalyInformation: http://school.diffi ety.org/page82/page82.html

25–28: VII Iberoamerican Conference on Topology and its Applications, Valencia, Spain Information: [email protected]; http://cita.webs.upv.es

30–July 3: Analysis, PDEs and Applications, Roma, ItalyInformation: [email protected]; http://www.mat.uniroma1.it/~mazya08/

28–July 1: 3rd Small Workshop on Operator Theory, Kra-kow, PolandInformation: [email protected];http://www.zzm.ar.krakow.pl/swot08/

30–July 3: Analysis, PDEs and Applications. On the occa-sion of the 70th birthday of V. Maz’ya, Roma, ItalyInformation: [email protected];http://www.mat.uniroma1.it/~mazya08/

30–July 4: Joint ICMI/IASE Study; Teaching Statistics in School Mathematics. Challenges for Teaching and Teach-er Education, Monterrey, MexicoInformation: [email protected]; http://www.ugr.es/~icmi/iase_study/

30–July 4: Geometry of complex manifolds, CIRM Luminy, Marseille, France Information: [email protected]; http://www.cirm.univ-mrs.fr

30–July 4: The European Consortium for Mathematics in Industry (ECMI2008), University College, London, UKInformation: [email protected]; http://www.ecmi2008.org/

30–July 5: 9th Conference on Geometry and Topology of Manifolds, Krakow, PolandInformation: [email protected]; http://www.im.uj.edu.pl/gtm2008/

July 2008

1–4: VIth Geometry Symposium, Bursa, Turkey Information: [email protected]; http://www20.uludag.edu.tr/~geomsymp/index.htm

2–4: The 2008 International Conference of Applied and Engineering Mathematics, London, UK Information: [email protected]; http://www.iaeng.org/WCE2008/ICAEM2008.html

2–11: Soria Summer School on Computational Mathemat-ics: “Algebraic Coding Theory” (S3CM), Universidad de Valladolid, Soria, Spain Information: [email protected]; http://www.ma.uva.es/~s3cm/

Conferences


3–8: 22nd International Conference on Operator Theory, West University of Timisoara, Timisoara, RomaniaInformation: [email protected]; http://www.imar.ro/~ot/

7–10: The Tenth International Conference on Integral Methods in Science and Engineering (IMSE 2008), Univer-sity of Cantabria, Santander, SpainInformation: [email protected], [email protected];http://www.imse08.unican.es/

7–10: International Workshop on Applied Probability (IWAP 2008), Compiègne, FranceInformation: [email protected],[email protected];http://www.lmac.utc.fr/IWAP2008/

7–11: VIII International Colloquium on Differential Geom-etry (E. Vidal Abascal Centennial Congress), Santiago de Compostela, Spain Information: [email protected]; http://xtsunxet.usc.es/icdg2008

7–11: Spectral and Scattering Theory for Quantum Mag-netic Systems, CIRM Luminy, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

7–11: Algebraic Topological Methods in Computer Sci-ence (ATMCS) III, Paris, FranceInformation: [email protected]; http://www.lix.polytechnique.fr/~sanjeevi/atmcs

7–11: Set Theory, Topology and Banach Spaces, Kielce, PolandInformation: [email protected]; http://www.pu.kielce.pl/~topoconf/

7–12: New Horizons in Toric Topology, Manchester, UKInformation: [email protected]; http://www.mims.manchester.ac.uk/events/workshops/NHTT08/

7–12: International Conference on Modules and Repre-sentation Theory, Cluj-Napoca, RomaniaInformation: [email protected], [email protected];http://math.ubbcluj.ro/~aga_team/AlgebraConferenceCluj 2008.html

13: Joint EWM/EMS Workshop, Amsterdam (The Nether-lands)Information: http://womenandmath.wordpress.com/joint-ewm ems-worskhop-amsterdam-july-13th-2007/

14–17: International Conference on Differential and Dif-ference Equations, Veszprem, HungaryInformation: [email protected]; http://www.szt.uni-pannon.hu/~ddea2008

14–18: Fifth European Congress of Mathematics (5ECM), Amsterdam, NetherlandsInformation: http://www.5ecm.nl

14–18: Effi cient Monte Carlo: From Variance Reduction to Combinatorial Optimization. A Conference on the Oc-casion of R.Y. Rubinstein’s 70th Birthday, Sandbjerg Estate, Sønderborg, DenmarkInformation: [email protected];http://www.thiele.au.dk/Rubinstein/

15–18: Mathematics of program construction, CIRM Lu-miny, Marseille, France Information: [email protected]; http://www.cirm.univ-mrs.fr

15–19: The 5th World Congress of the Bachelier Finance Society, London, UKInformation: mark@chartfi eld.org; http://www.bfs2008.com

17–19: 7th International Conference on Retrial Queues (7th

WRQ), Athens, GreeceInformation: [email protected]; http://users.uoa.gr/~aeconom/7thWRQ_Initial.html

17–August 1: XI Edition of the Italian Diffi ety School, Santo Stefano del Sole (Avellino), ItalyInformation: http://school.diffi ety.org/page3/page0/page64/page64.html

18–22 : European Open Forum ESOF 2008: Science for a better life, Barcelona, SpainInformation: http://www.esof2008.org

20–23: International Symposium on Symbolic and Alge-braic Computation, Hagenberg, AustriaInformation: [email protected]; http://www.risc.uni-linz.ac.at/about/conferences/issac2008/

21–24: SIAM Conference on Nonlinear Waves and Coher-ent Structures, Rome, ItalyInformation: [email protected]; http://www.siam.org/meetings/nw08/

21–25: Operator Structures and Dynamical Systems, Lei-den, the NetherlandsInformation: [email protected];http://www.lorentzcenter.nl/lc/web/2008/288/info.php3?wsid =288

21–25: Summer School “PDE from Geometry”, Cologne, GermanyInformation: [email protected];http://www.mi.uni-koeln.de/~gk/school08

21–August 29: CEMRACS, CIRM Luminy, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

24–26: Workshop on Current Trends and Challenges in Model Selection and Related Areas, University of Vienna, Vienna, AustriaInformation: [email protected]; http://www.univie.ac.at/workshop_modelselection/

August 2008

3–9: Junior Mathematical Congress, Jena, GermanyInformation: [email protected]; http://www.jmc2008.org/

13–19: XXVII International Colloquium on Group Theoreti-cal Methods in Physics (Group27), Yerevan, ArmeniaInformation: [email protected]; http://theor.jinr.ru/~group27

14–19: Complex Analysis and Related Topics (The XIth Ro-manian-Finnish Seminar), Alba Iulia, RomaniaInformation: rofi [email protected];http://www.imar.ro/~purice/conferences/Ro-Fin-11/rofin11 sem.html

Conferences


16–31: EMS-SMI Summer School: Mathematical and nu-merical methods for the cardiovascular system, Cortona, ItalyInformation: [email protected]

18–22: International conference on ring and module the-ory, Ankara, TurkeyInformation: http://www.algebra2008.hacettepe.edu.tr

19–22: Duality and Involutions in Representation Theory, National University of Ireland, Maynooth, IrelandInformation: [email protected];http://www.maths.nuim.ie/conference/

21–23: International Congress of 20th Jangjeon Mathemat-ical Society, Bursa, TurkeyInformation: [email protected], [email protected], [email protected]; http://www20.uludag.edu.tr/~icjms20/

25–28: The 14th general meeting of European Women in Mathematics (EWM), Novi Sad, SerbiaInformation: [email protected];http://ewm2009.wordpress.com/

25–29: Function spaces, Differential Operators and Non-linear Analysis, Helsinki, FinlandInformation: [email protected] ; http://mathstat.helsinki.fi /fsdo-na

27–29: Journées MAS de la SMAI: Modélisation et Statis-tiques des Réseaux, Rennes, FranceInformation: [email protected];http://mas2008.univ-rennes1.fr/

28–September 2: 9ème Colloque Franco-Roumain de Mathématiques Appliquées, Brașov, RomaniaInformation: [email protected];http://cs.unitbv.ro/colloque2008/site/

29–September 2: The International Conference of Differ-ential Geometry and Dynamical Systems (DGDS-2008) and The V-th International Colloquium of Mathematics in Engineering and Numerical Physics (MENP-5, mathemat-ics sections), Mangalia, Romania Information: [email protected]; [email protected]://www.mathem.pub.ro

31–September 5: The 22nd Summer Conference on Real Functions Theory, Stara Lesna, SlovakiaInformation: http://www.saske.sk/MI/confer/lsrf08.html

31–September 6: Summer School on General Algebra and Ordered Sets, Trest, Czech RepublicInformation: http://www.karlin.mff.cuni.cz/~ssaos

September 2008

1–5: Representation of surface groups, CIRM Luminy, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

1–5: Conference on Numerical Analysis (NumAn 2008), Kalamata, GreeceInformation: [email protected]; http://www.math.upatras.gr/numan2008/

1–6: School (and Workshop) on the Geometry of Algebraic Stacks, Trento, Italy

Information: [email protected]; http://www.science.unitn.it/cirm/

1–12: School on Algebraic Topics of Automata, Lisbon, PortugalInformation: [email protected]; http://caul.cii.fc.ul.pt/SATA2008/

2–5: X Spanish Meeting on Cryptology and Information Security, Salamanca, SpainInformation: [email protected]; http://www.usal.es/xrecsi/english/main.htm

2–7: International Conference on Geometry, Dynamics, Integrable Systems, Belgrade, SerbiaInformation: [email protected]; http://www.mi.sanu.ac.yu/~gdis08/

6–14: First European Summer School in Mathematical Fi-nance, Dourdan near Paris, FranceInformation: euroschoolmathfi @cmap.polytechnique.fr; http://www.ceremade.dauphine.fr/~bouchard/ESCMF/

8–12: Chinese-French meeting in probability and analysis, CIRM Luminy, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

8–10: Calculus of Variations and its Applications From Engineering to Economy, Universidade Nova de Lisboa, Ca-parica, PortugalInformation: [email protected]; http://ferrari.dmat.fct.unl.pt/cva2008/

8–12: International Workshop on Orthogonal Polynomi-als and Approximation Theory 2008. Conference in honor of professor Guillermo López Lagomasino’s 60th birthday, Universidad Carlos III de Madrid, Leganés, SpainInformation: [email protected]; http://www.uc3m.es/iwopa08

8–19: EMS Summer School: Mathematical models in the manufacturing of glass, polymers and textiles, Montecati-ni, ItalyInformation: [email protected] .it; http://web.math.unifi .it/users/cime//

9–12: INDAM Workshop on Holomorphic Iteration, Semi-groups and Loewner Chains, Rome, ItalyInformation: [email protected];http://www.congreso.us.es/holomorphic/

10–12: Nonlinear Differential Equations (A Tribute to the work of Patrick Habets and Jean Mawhin on the occasion of their 65th birthdays), Brussels, BelgiumInformation: [email protected]; http://www.uclouvain.be/node2008.html

12–15: Mathematical Analysis, Differential Equations and Their Applications, Famagusta, North CyprusInformation: [email protected];http://madd2008.emu.edu.tr/contact.php

14–18: 7th Euromech Fluid Mechanics Conference, Man-chester, UKInformation: http://www.mims.manchester.ac.uk/events/work-shops/EFMC7/

Conferences


15–19: Geometry and Integrability in Mathematical Phys-ics, CIRM Luminy, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

15–19: International Conference on K-Theory and Homo-topy Theory, Santiago de Compostela, SpainInformation: [email protected]; http://www.usc.es/regaca/ktht/15–19: 9th SIMAI Congress, Rome, ItalyInformation: [email protected]; http://www.simai.eu/

16–20: International Conference of Numerical Analysis and Applied Mathematics 2008 (ICNAAM 2008), Psalidi, Kos, GreeceInformation: [email protected]; http://www.icnaam.org/

18–21: 6th International Conference on Applied Mathemat-ics (ICAM6), Baia Mare, RomaniaInformation: [email protected]; http://www.icam.ubm.ro

18–29: Crimean Autumn Mathematical School-Sympo-sium, Batiliman, UkraineInformation: [email protected]

19–26: International Conference on Harmonic Analysis and Approximations IV, Tsaghkadzor, ArmeniaInformation: [email protected]; http://math.sci.am/conference/sept2008/conf.html

21–24: The 8th International FLINS Conference on Compu-tational Intelligence in Decision and Control (FLINS 2008), Madrid, SpainInformation: fl [email protected]; http://www.mat.ucm.es/congresos/fl ins2008

22–25: Symposium on Trends in Applications of Mathe-matics to Mechanics (STAMM 2008), Levico, ItalyInformation: [email protected]; http://mate.unipv.it/pier/stamm08.html

22–26: 10th International workshop in set theory, CIRM Lu-miny, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

22–28: 4th International Kyiv Conference on Analytic Number Theory and Spatial Tessellations, Jointly with 5th

Annual International Conference on Voronoi Diagrams in Science and Engineering (dedicated to the centenary of Georgiy Voronoi), Kyiv, UkraineInformation: [email protected]; http://www.imath.kiev.ua/~voronoi

29–October 3: Commutative algebra and its interactions with algebraic geometry, CIRM Luminy, Marseille, France Information: [email protected]; http://www.cirm.univ-mrs.fr

29–October 8: EMS Summer School: Risk theory and re-lated topics, Będlewo, PolandInformation: [email protected]; www.impan.gov.pl/EMSsummerSchool/

October 2008

3–5: II Iberian Mathematical Meeting, Badajoz, SpainInformation: [email protected]; http://imm2.unex.es

5–12: International Conference “Differential Equations. Function Spaces. Approximation Theory” dedicated to the 100th anniversary of the birthday of S.L. Sobolev, Nov-osibirsk, RussiaInformation: [email protected]; http://www.math.nsc.ru/conference/sobolev100/english

6–10: Partial differential equations and differential Galois theory, CIRM Luminy, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

9–11: Algebra, Geometry and Mathematical Physics, Tartu, EstoniaInformation: [email protected]; http://www.agmf.astralgo.eu/tartu08/

9–12: The XVI-th Conference on Applied and Industrial Mathematics (CAIM 2008), Oradea, RomaniaInformation: [email protected]; http://www.romai.ro

13–17: Hecke algebras, groups and geometry, CIRM Lu-miny, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

20–24: Symbolic computation days, CIRM Luminy, Mar-seille, France Information: [email protected]; http://www.cirm.univ-mrs.fr

22–24: International Conference on Modeling, Simulation and Control 2008, San Francisco, USA Information: [email protected]; http://www.iaeng.org/WCECS2008/ICMSC2008.html

26–28: 10th WSEAS Int. Conf. on Mathematical Methods and Computational Techniques in Electrical Engineering (MMACTEE 8), Corfu, GreeceInformation: [email protected]; http://www.wseas.org/conferences/2008/corfu/mmactee/

26–28: 7th WSEAS Int. Conf. on Non-Linear Analysis, Non-Linear Systems and Chaos (NOLASC 8), Corfu, GreeceInformation: [email protected]; http://www.wseas.org/conferences/2008/corfu/nolasc/

26–31: New trends for modeling laser-matter interaction, CIRM Luminy, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

27–31: New trends for modeling laser-matter interaction, CIRM Luminy, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

November 2008

3–7: Harmonic analysis, operator algebras and represen-tations, CIRM Luminy, Marseille, France Information: [email protected]; http://www.cirm.univ-mrs.fr

5–7: Fractional Differentiation and its Applications, An-kara, Turkey Information: [email protected]; http://www.cankaya.edu.tr/fda08/

Recent books


Recent Booksedited by Ivan Netuka and Vladimír Soucek (Prague)

Books submitted for review should be sent to: Ivan Netuka, MÚUK, Sokolovská, 83, 186 75 Praha 8, Czech Republic.

C. D. Aliprantis, R. Tourky: Cones and Duality, Graduate Studies in Mathematics, vol. 84, American Mathematical Society, Providence, 2007, 279 pp., USD 55, ISBN 978-0-8218-4146-4 Ordered vector spaces and cones were introduced in mathemat-ics at the beginning of the 20th century and were developed in parallel with functional analysis and operator theory. Since cones are employed to solve optimization problems, the theory of ordered vector spaces is an indispensable tool for solving a variety of applied problems appearing in areas such as engineer-ing, econometrics and the social sciences.

The aim of this book is to present the theory of ordered vector spaces from a contemporary perspective, which has been infl uenced by a study of ordered vector spaces in econom-ics as well as other recent applications. The material is spread out in eight chapters. The fi rst chapters start with fundamen-tal properties of wedges and cones and illustrate a variety of remarkable results from the connection between the topology and the order. The role of the Riesz decomposition property and normal cones is pointed out. The next section studies in detail cones in fi nite dimensional spaces and polyhedral cones. The authors proceed with a presentation of the fi xed points and eigenvalues of Krein operators. Further chapters contain mate-rial on K-lattices, Riesz-Kantorovich functionals and piecewise affi ne functions, which is a topic that has not been included in any monograph before. The last chapter serves as an appendix on linear topologies and their basic properties.

At the end of each section, there is a list of exercises of varying degrees of diffi culty designed to help the reader to better understand the material. This aim is further supported by the provision of hints to selected exercises. Since the top-ics discussed in the book have their origins in problems from economics and fi nance, the book will be valuable not only for students and researchers in mathematics but also for those in-terested in economics, fi nance and engineering. (jsp)

J. Appell, M. Väth: Elemente der Funktionalanalysis, Vieweg, Wiesbaden, 2005, 349 pp., EUR 24,90, ISBN 3-528-03222-7 This book provides an elementary introduction to both linear and nonlinear functional analysis. The authors emphasize the variability of infi nite dimensional spaces and highlight the role of compactness. The linear part of the book covers classical means of doing functional analysis (normed linear spaces and opera-tors, the criteria of compactness in varied spaces and the Riesz-Schauder theory of compact operators). The second part of the book concentrates mainly on fi xed point theorems (Banach, Brouwer, Schauder, Darbo, Borsuk). The appendix includes the Baire category theorem, bases in Banach spaces, the Weierstrass approximation theorem and the Tietze-Urysohn and Dugundji extension theorems.

Each chapter ends with many exercises and problems of varying diffi culty, which give further applications and exten-

5–7: Modern Problems of Differential Geometry and Gen-eral Algebra, Saratov City, RussiaInformation: [email protected];http://mexmat.sgu.ru/vagner2008.php?lang=en

10–14: The 6th Euro-Maghreb workshop on semigroup theory, evolution equations and applications, CIRM Lu-miny, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

17–21: Geometry and topology in low dimension, CIRM Luminy, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

21–22: International Conference on Nolinear Analysis and Applied Mathematics, Targoviste, RomaniaInformation: [email protected]; http://icnaam.valahia.ro/

24–28: Approximation, geometric modelling and applica-tions, CIRM Luminy, Marseille, FranceInformation: [email protected]; http://www.cirm.univ-mrs.fr

December 2008

1–5: Homology of algebra: structures and applications, CIRM Luminy, Marseille, Franceinformation: [email protected]; http://www.cirm.univ-mrs.fr

8–12: Latent variables and mixture models, CIRM Luminy, Marseille, Franceinformation: [email protected]; http://www.cirm.univ-mrs.fr

15–19: Meeting on mathematical statistics, CIRM Luminy, Marseille, Franceinformation: [email protected]; http://www.cirm.univ-mrs.fr

16–18: Eighth IMA International Conference on Mathe-matics in Signal Processing, Cirencester, UKInformation: [email protected]; http://www.ima.org.uk/Conferences/signal_processing/signal_processing08.html

February 2009

5-8 : European Student Conference EUROMATH 2009, Cy-prusInformation: www.euromath.org

March 2009

15-20: ALGORITMY 2009 – Conference on Scientifi c Com-puting, High Tatra Mountains, Podbanske, SlovakiaInformation: [email protected]; http://www.math.sk/alg2009

July 2009

6–10: 26th Journées arithmétiques, Saint-Etienne, FranceInformation: [email protected]; http://ja2009.univ-st-etienne.fr

Recent books


sions of the theory. The book is easily understandable and it can be warmly recommended to graduate students in mathematics, physics, biology, chemistry and engineering as a neat introduc-tion to functional analysis. (jl)

G. Balkerma, P. Embrechts: High Risk Scenarios and Ex-tremes, Zurich Lectures in Advanced Mathematics, European Mathematical Society, Zurich, 2007, 375 pp., EUR 48, ISBN 978-3-03719-035-7 The problem of multivariate extremes and their fi nancial appli-cation is of central interest in the fi eld of quantitative risk man-agement and it is the main topic of this monograph. The book is divided into fi ve parts and twenty chapters. After a description of motivations, the fi rst part contains an introduction to the the-ory of point processes and their applications in extreme value theory. The second part covers the classical univariate and mul-tivariate extreme value theories. The coordinate-wise approach and max-stable distributions are discussed here.

The main part of the book comprises parts III and IV. In the third part a modern geometric (coordinate free) approach to multivariate extremes is broadly studied. Particular interest is paid to heavy tailed distributions and to the generalized Pareto distribution (candidate multivariate extension of the GPD). Extreme value theory is often based on threshold exceedance, which is studied in the fourth part of the book. There is no unique threshold in the multivariate case and the authors study two classes (horizontal and elliptic) of thresholds. In the last part of the book, some open problems in the theory and statisti-cal applications of multivariate extremes are summarized.

The book is written with a broad audience of theoretical and applied mathematicians in mind. Problems are mostly motivated by examples from insurance and fi nance (portfolio theory) but their solution is purely mathematical. The level of rigour in the book is very high and to understand all the tech-niques of modern extreme value theory, a solid mathematical background is required. On the other hand, if one just needs to apply the results, the proofs and technical sections may be skipped. The book can also be recommended as the basis for an advanced course on multivariate extreme value theory. (dh)

W. Byers: How Mathematicians Think. Using Ambiguity, Contradiction, and Paradox to Create Mathematics, Princeton University Press, Princeton, 2007, 415 pp., USD 35, ISBN 978-0-691-12738-5 A lot has been said and written about the philosophy of math-ematics, yet not enough. There are too many unanswered ques-tions. For example, is mathematics discovered or created? There still seems to be room for a good book in this fi eld, and this one is wonderful, a must-read for everyone interested in mathemat-ics, philosophy and/or history. One of the most pervasive myths about mathematics is that it is a dull technocratic discipline car-ried out by grim computer-like minds that have no feelings and who work without intuition, ambiguity or doubt and produce strictly formal algorithms and theorems free of contradictions, confl icts and paradoxes. Such myths, unfortunately still rather common among ‘outsiders’, call for such a book.

The author, a great mathematician and philosopher, and also a practitioner of Zen-Buddhism, shows how essential non-logical qualities are in mathematical research and creativity and that the secret of successful mathematics is not in its logical

structure, or at least not only there. Excellent discussions are presented about ambiguity, contradiction, paradox and their central role in the world of mathematical discovery. (lp)

A. Carbery et al., Eds.: Complex and Harmonic Analysis, Pro-ceedings, May 25-27, 2006, Thessaloniki, DEStech Publications, Lancaster, 2007, 327 pp., USD 89,50, ISBN 978-1-932078-73-2This book represents the proceedings of the international con-ference on complex and harmonic analysis, which was held in May 2006 at the Aristotle University of Thessaloniki. The vol-ume contains 23 articles from a broad range of topics including geometric function theory, complex iteration, function spaces, composition operators, extremal problems, potential theory, maximal functions, analysis on manifolds and others. The con-ference was dedicated to the memory of Nikos Danikas (a member of Thessaloniki University) who died in 2004. The in-troductory paper on the mathematical work of Nikos Danikas is written by W.K. Hayman and it is followed by an article on Danikas measures written by V. Nestoridis. The book can be recommended to students and researchers in the area. (jl)

A. D. D. Craik: Mr Hopkins’ Men. Cambridge Reform and British Mathematics in the 19th Century, Springer, Berlin, 2007, 405 pp., 78 fi g., EUR 99.95, ISBN 978-1-84628-790-9 This book gives a fascinating view of Cambridge University during the Victorian era. It is divided into two parts. The fi rst part “Educating the Elite” describes the system of education in Cambridge, its dramatic changes in the 19th century and its role in the development of the mathematical sciences in Britain. The author describes the evolution of curricula and reforms lead-ing to Cambridge’s dominance of British higher education. The Cambridge reforms are analysed in a comprehensive context (the role of political decisions, the infl uence of the church and parliament, the development of the town and colleges in Cam-bridge, life at the university, social interest in higher education and scientifi c work, and so on). A detailed biography of William Hopkins, the most remarkable tutor in Cambridge in the mid-19th century, and an appreciation of his mathematical achieve-ments have been created with the help of a study of archival sources. Brief biographies of Hopkins’ top students from 1829 up to 1854 are added at the end of the fi rst part. Pencil and wa-tercolour portraits of top students from Hopkins’ own collec-tion (attributed to the artist T. C. Wageman, who created them between 1829 and 1852) are published here for the fi rst time.

The second part “Careers of the Wrangles” is devoted to a description of the careers of some top students and gradu-ates (so-called Wrangles) including many famous scientists and mathematicians, professors at English and Scottish universi-ties and colleges, fi rst professors in Australia, offi cial educators overseas and throughout the colonies, tutors of prominent peo-ple (for example an Indian maharajah), churchmen, etc. De-tailed biographies of four excellent scientists (G. Green, J. C. Adams, G. G. Stokes and H. Goodwin) are included. At the end of the second part, the author describes the transformation of Cambridge from an unimportant institution into a world centre for mathematical and physical sciences and education. Various topics are discussed (growth of a research community, the birth of journals, the development of scientifi c institutions, the analy-sis of achievements in mathematical sciences before 1830 and then from 1830 up to 1880).

Recent books


The book can be recommended to people who are interest-ed in the history of Victorian Britain in general and in the his-tory of Cambridge University, mathematical education, math-ematics, and scientifi c life and work, as well as the connections of science and religious belief, politics, etc. (mbec)

P. Daskalopoulos, C.E. Kenig: Degenerate Diffusions. Initial Value Problems and Local Regularity Theory, Tracts in Math-ematics, vol. 1, European Mathematical Society, Zürich, 2007, 198 pp., EUR 48, ISBN 978-3-03719-033-3 This book is devoted to the study of nonnegative solutions of the Cauchy initial value problem or the Dirichlet boundary val-ue problem for a class of nonlinear evolution differential equa-tions ∂u/∂t = ∆ ϕ(u) on Rn x (0,T) or on a cylinder B x (t1,t2), where the nonlinearity ϕ is assumed to be continuous and in-creasing with ϕ (0) = 0, and is assumed to satisfy the growth condition a ≤ [u ϕ ´(u)]/[ϕ (u)] ≤ 1/a for all u > 0 for a in (0,1) and the normalization condition ϕ (1) = 1. This class forms a natural generalization of the power case ϕ(u) = um and has a wide range of applications both in physics and in geometry. In 1940, D. Widder characterized the set of all nonnegative weak solutions of the Cauchy problem for the heat equation on the strip Rn x (0,T) by fi ve fundamental properties.

The authors are interested in analogues of these properties for the nonlinearities ϕ described above. The results are divided into two groups according to the growth of ϕ. The fi rst group concerns the case of “slow diffusion” and it is characterized in the power case by the condition m > 1. The second group includes the supercritical “fast diffusion” case, which general-izes the case ϕ (u) = um for (n-2)/n < m < 1. Besides the items mentioned above, the book collects together local regularity results in chapter 1 (a priori L∞ bounds, the Harnack inequality and equicontinuity of solutions to the slow diffusion case) and a proof of continuity of weak solutions to the porous medium equation in the fi nal (fi fth) chapter.

The book gives an up-to-date, clear and concise overview of results concerning degenerate diffusion together with power-ful methods and useful techniques for studying existence and qualitative properties of solutions. A number of comments and discussions of various topics, a brief summary of further known results and some open problems are listed in the last section of each chapter and an up-to-date bibliography will be appreci-ated by both researchers and graduate students with a back-ground in analysis and partial differential equations. (jsta)

P. Drábek, J. Milota: Methods of Nonlinear Analysis. Appli-cations to Differential Equations, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2007, 568 pp., EUR 74.79, ISBN 978-3-7643-8146-2 In many cases, existence problems in the theory of differential equations and in the calculus of variations refer to results of nonlinear analysis in abstract infi nite-dimensional spaces. For example, the problem of fi nding stationary points of a function-al in the calculus of variations may lead to an abstract saddle point theorem. To prove existence of a solution of a problem in the theory of partial differential equations, fi xed points and other topological tools of nonlinear operator theory are often useful. The book shows methods of how to apply abstract non-linear analysis to specifi c problems. Thus both nonlinear func-tional analysis and application sections are well developed.

The list of all the topics would be too long so we will only mention the main themes. The exposition of differential cal-culus in normed linear spaces discusses the inverse function theorem and the implicit function theorem. Some further top-ics are specifi c to fi nite dimensions; they include the rank theo-rem, fi nite-dimensional bifurcation theorems, integration of differential forms on manifolds with Stokes’ theorem and the Brouwer degree. The topological methods in nonlinear analysis are based on infi nite-dimensional generalizations of the degree. The Leray-Schauder degree is available for compact perturba-tions of the identity operator. Another extension of the concept of degree can be used for monotone operators and their gen-eralizations. Applications include fi xed point theorems, global bifurcation theorems and existence theorems based on subso-lutions and supersolutions.

The exposition of variational methods starts with a classical analysis of extrema on infi nite dimensional spaces and direct methods of the calculus of variations. The analysis of stationary points covers bifurcation of potential operators, the mountain pass lemma, the Lusternik-Schnirelmann method and other tools for searching saddle points. Finally, the last chapter is de-voted to a systematic treatment of classical and weak solutions of partial differential equations, demonstrating techniques developed in the preceding text. The authors reveal the fruits of their long and rich teaching experience. The presentation is self-contained and covers all the fundamental tools of nonlin-ear analysis. On the other hand, they also organize advanced excursions, mostly as appendices, to topics of interest and some parts are developed up to recent research level. Consequently, this volume can be used as a reference book or a textbook, es-pecially as an important source of knowledge and inspiration for university students and teachers of all levels. (jama)

H. Dym: Linear Algebra in Action, Graduate Studies in Math-ematics, vol. 78, American Mathematical Society, Providence, 2007, 541 pp., USD 79, ISBN 978-0-8218-3813-6 This is a book on linear algebra, playing an important role in pure and applied mathematics, computer science, physics and engineering. The book is divided into 23 chapters and 2 appen-dices. The fi rst six chapters and some selected parts from chap-ters 7-9 are based on classical linear algebra topics. The reader will fi nd here many interesting principles and results concern-ing vector spaces, Gaussian elimination and its applications, determinants, eigenvalues and eigenvectors, Jordan forms and their calculations, normed linear spaces, inner product spaces and orthogonality, and symmetric, Hermitian and normal ma-trices. The next three chapters are devoted to singular values and related inequalities, pseudoinverses and triangular factori-zation and positive defi nite matrices.

Chapter 13 treats difference equations, differential equa-tions and their systems. Chapters 14-16 contain applications to vector valued functions, the implicit function theorem and ex-tremal problems. The subsequent chapters deal with matrix val-ued holomorphic functions, matrix equations, realization theory, eigenvalue location problems, zero location problems, convexity and matrices with nonnegative entries. Two appendices describe useful facts from complex function theory. The book offers ba-sic and advanced techniques of linear algebra from the point of view of analysis. Each technique is illustrated by a wide sam-ple of applications and it is accompanied by many exercises of

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varying diffi culty, which give further extensions of the theory. The book can be recommended as a general text for a variety of courses on linear algebra and its applications, as well as a self-study aid for graduate and undergraduate students. (mbec)

H.-D. Ebbinghaus: Ernst Zermelo – An Approach to His Life and Work, Springer, Berlin, 2007, 356 pp., EUR 49.95, ISBN 978-3-540-49551-2 This book gives the fi rst English, detailed scientifi c biography of Ernst Zermelo (1871–1953), a German mathematician best known for a formulation of the axiom of choice and for an axi-omatization of set theory. He is considered to be one of the greatest logicians in the fi rst half of the 20th century. In fi ve chapters, the author describes Zermelo’s life and his scientifi c achievements from his youth through his studies, works, teach-ing and scientifi c activities. The author also analyses Zermelo’s conditions for scientifi c study and research before, during and after World War II, his isolation during the war and his activi-ties after the war. Zermelo’s major scientifi c contributions (in set theory, the calculus of variations, the theory of games, the theory of rating systems, applied mathematics) are discussed without assuming a detailed mathematical knowledge of the fi eld in question.

The presentation of Zermelo’s works explores his motiva-tions, aims, acceptance and infl uence on the development of mathematics. The book is based on a study of unknown and unpublished sources, archival materials, private letters, family documents, previously unpublished notes, interviews and mem-oirs. The main text is followed by a schematic curriculum vitae of Ernst Zermelo summarizing the most important events of his life. The book concludes with an appendix containing a Ger-man version of Zermelo’s unpublished ideas and parts of his works (for example selected proofs), his original letters, notes and samples of his literary activities. The book contains more than 40 photos and facsimiles, a large list of references and an index. It gives new light to all facets of Zermelo’s life and math-ematical achievements and it can be recommended to a wide audience. The book is suitable for mathematicians, historians of mathematics and science, students and teachers. (mbec)

J. Friberg: A Remarkable Collection of Babylonian Math-ematical Texts, Sources and Studies in the History of Mathemat-ics and Physical Sciences, Springer, Berlin, 2007, 533 pp., EUR 99.95, ISBN 978-0-387-34543-7 This fascinating book presents 121 unpublished mathematical clay tablets from the Norwegian Schøyen Collection includ-ing many interesting tablets from the Early Dynastic III pe-riod (circa 2600–2350 BC) up to the Late Kassite period (circa 14th–13th century BC). Based on descriptions, transliterations, translations, interpretations, analysis and synthesis of the math-ematical cuneiform texts and their comparison with previously published mathematical clay tablets, the author has made nu-merous amazing discoveries and has created a new comprehen-sive treatment of Old Babylonian mathematical texts. The book is divided into 12 chapters, 10 appendices, a vocabulary for MS texts, an index of subjects, an index of texts and a large list of references.

The author starts with an introduction explaining how to better understand mathematical cuneiform texts. He explains mathematical terminology, cuneiform systems of notations for

numbers and measures, operations with sexadecimal numbers (addition, subtraction, multiplication, division, and an ingen-ious factorization method for the computation of reciprocals and square roots). Then he explains the use of the Old Baby-lonian standard tablets of squares, square roots, cube roots and reciprocals, as well as the non-standard tablets of “quasi-cube roots”. The next chapters are devoted to a discussion of Old Babylonian metrological systems (standard types for volumes, area, length and weight are described).

A large part of the book analyses some typical Babylonian geometrical problems (computation of the area of a triangle, a trapezoid or a circle; computation of the diagonals of the sides of a square; the problem of how to inscribe and circumscribe circles, triangles and squares, how to divide fi gures into other fi gures with given special properties, and how to change fi gures to other fi gures, while retaining the area; computation of the volume of prisms, cubes and cylinders; construction of laby-rinths, mazes, decorative patterns and ground plans of palaces), some typical algebraic problems (arithmetic and geometric progressions, solutions of linear and quadratic equations and their systems) and some problems from daily life (construction of walls, building of a ramp, dividing works, money and so on). Many pictures, drawings and coloured photos of the most inter-esting tablets are also included. The book opens up Babylonian mathematics to a new generation of mathematicians, historians of science and mathematics, teachers and students. It can there-fore be recommended to a wide audience. (mbec)

B. Fristedt, N. Jain, N. Krylov: Filtering and Prediction – A Primer, Student Mathematical Library, vol. 38, American Math-ematical Society, Providence, 2007, 252 pp., USD 39, ISBN 978-0-8218-4333-8 This book is mainly intended as an introduction to the prob-lem of fi ltering and prediction in time homogeneous Markov chains and the Wiener process. The problem of how to estimate an unobservable signal based on an available observation of re-sponse (a signal corrupted by noise) is known as fi ltering. Pre-diction means estimating the future of a random process based on its history.

The fi rst two chapters of the book are introductory and deal with basic probability concepts and discrete Markov chains. In the third chapter, the discrete Markov chains are used as the simplest model for an introduction of the fi ltering problem. In chapter 4, the conditional expectation is introduced as a neces-sary tool for continuous (in space and/or time) processes. The continuous space Markov chains are then discussed in chapter 5; the famous Kalman fi lter is described and the chapter closes with linear fi ltering. Chapter 6 covers fi ltering for the Wiener process and the continuous time Kalman fi lter is introduced in a classical setting. The last two chapters deal with the prediction problem in stationary random sequences.

The book is written in an elementary way (no special knowledge of probability and statistics is needed to read the book) but it is still mathematically rigorous. The book can be recommended to all students interested in stochastic models. Since the book contains many problems and remarks, it can also be recommended as the basis of a one semester introduc-tory course; almost all theorems are proved and problems may be used as homework assignments although some of the prob-lems are more challenging than others. (dh)

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M. Giaquinta, G. Modica: Mathematical Analysis – Linear and Metric Structures and Continuity, Birkhäuser, Boston, 2007, 465 pp., EUR 119, ISBN 978-0-8176-4374-4 This book provides a comprehensive explanation of the back-ground of functional analysis and its origins. It is divided into three parts. It begins with a careful and inspiring exposition of basic ideas on linear and metric structures. The second part is dedicated to the role of general topological notions in the framework of metric spaces. These two parts help the reader to prepare for the main principles of functional analysis, which are then explained in the third part of the book. At each step, theory is accompanied by important examples and applications. Carefully chosen exercises stimulate the reader to work actively with the text. On the whole, this self-contained and up-to-date book can be very useful not only to the advanced undergradu-ate or graduate student but also to anybody who wants to sys-temize their knowledge on the elements of the subject. (oj)

J. J. Gray, K. H. Parshall, Eds.: Episodes in the History of Modern Algebra (1800–1950), History of Mathematics, vol. 32, American Mathematical Society, Providence, 2007, 336 pp., USD 69, ISBN 978-0-8218-4343-7 This book offers new light on the development and history of modern algebra. The book brings together suitably revised, chapter-length versions of twelve lectures that were given at the workshop on the history of 19th and 20th century algebra held at the Mathematical Sciences Research Institute in Ber-keley in 2003.

The introduction and chapters written by prominent math-ematicians and historians of mathematics (J. J. Gray, K. H. Parshall, E. L. Ortiz, S. E. Despeaux, O. Neumann, H. M. Ed-wards, G. Frei, J. Schwermer, D. D. Fenster, Ch. W. Curtis, L. Corry, N. Schappacher, S. Slembek, C. McLarty) provide complex and detailed overviews of the evolution of modern algebra from the early 19th century work of Ch. Babbage on function equations through to the description of the develop-ment of calculus operations (British research done before and documented within the Cambridge Mathematical Journal), the analysis of divisibility theories in the early history of com-mutative algebra (contributions by C. F. Gauss, E. E. Kummer, E. I. Zolotarev, L. Kronecker , D. Hilbert, E. Noether, etc.), a presentation of Kronecker’s fundamental theorem on general arithmetic, a description of advances in the theory of algebras and algebraic number theory (works by J. H. M. Wedderburn, A. Hurwitz, R. Brauer, H. Hasse, E. Noether, H. Minkowski, K. Hensel, L. Dickson, A. A. Albert, etc.) and in the analysis of the foundations of algebraic geometry and its arithmetization and development (results of H. Hasse, E. Noether, F. Severi, A. Grothendieck, etc.).

The topics discussed represent the long and diffi cult process of the changing organization of the main subjects, the chang-ing algebraic themes, ideas and concepts, as well as the results of mathematical communication and collaboration within and across national boundaries. A comprehensive list of references as well as many detailed notes are added at the end of each chapter. The book can be recommended to readers who are in-terested in the history of modern mathematics in general and modern algebra in particular. It is suitable for mathematicians, historians of mathematics and science, teachers and students. (mbec)

M. Greenacre: Correspondence Analysis in Practice, 2nd edi-tion, Chapman and Hall/CRC, Boca Raton, 2007, USD 79,95, ISBN 1-58488-616-1The previous edition, published in 1993, has been totally re-vised and extended. Like the fi rst edition, this book is intended to be practically oriented and didactic. The author’s wish is to represent in each of the 25 chapter (and fi ve appendices) a fi xed amount of material. As a result, each chapter is exactly eight pages in length to offer a fi xed amount of material both for reading and teaching. However, this does not always have pleasant consequences in some chapters, where more informa-tion would be useful for the reader. A short summary of the main points in the form of a list concludes each chapter.

As in the fi rst edition, the book’s main thrust is toward the practice of correspondence analysis, so that most technical is-sues and mathematical aspects are gathered in appendix A. The theoretical part here is more extensive than that of the fi rst edi-tion, including additional theory on new topics such as canoni-cal correspondence analysis, transition and regression relation-ships, stacked tables, subset correspondence analysis and the analysis of square tables. Concerning computational issues, a very strong feature of this edition is the use of the R program for all computations. The lengthy appendix B (45 pages) sum-marizes (on examples) possibilities of relating macros from R. In addition to that, three different technologies are described enabling the creation of the graphical displays presented in the book. No references at all are given in the 25 chapters; a rela-tively brief bibliography in appendix C is given to point readers toward further reading containing a more complete literature guide. A glossary of the most important terms and an epilogue with some fi nal thoughts conclude the book.

Summarizing, this is a nice book for all those who wish to acquaint themselves with a versatile methodology of corre-spondence analysis and the way it can be used for the analysis and visualization of data arriving typically from the fi elds of social, environmental and health sciences, marketing and eco-nomics. Numerous examples provide a real fl avour of the pos-sibilities of the method. (jant)

D. D. Haroske, H. Triebel: Distributions, Sobolev Spaces, El-liptic Equations, EMS Textbooks in Mathematics, European Mathematical Society, Zürich, 2007, 294 pp., EUR 48, ISBN 978-3-03719-042-5 Many problems in mathematical physics reduce to elliptic dif-ferential equations of second order and their boundary value problems, and also to the spectral theory of such operators. Central role are played by the Dirichlet problem, the Neumann problem and the eigenvalue and eigenfunction problem. The main aim of the book is to develop the L2 theory of the listed problems on bounded domains with smooth boundaries in Eu-clidean space. This is done in a reader-friendly way at a mod-erate level of diffi culty, aiming at graduate students and their teachers.

Among the topics covered, we fi nd the classical theory of the Laplace-Poisson equation, the theory of distributions, the theory of Sobolev spaces on domains, abstract spectral theory in Hilbert and Banach spaces and compact embeddings. One of the principal assets of the book is a very good, friendly and accessible introduction to various aspects of function space the-ory and their applications in the theory of partial differential

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equations. The book is richly furnished with interesting exercis-es and a thorough set of notes is attached to each chapter. This is an ideal book for both students and teachers of a modern graduate course on partial differential equations. (lp)

M. Hata: Problems and Solutions in Real Analysis, Series on Number Theory and Its Applications, vol. 4, World Scientifi c, New Jersey, 2007, 292 pp., USD 42, ISBN 978-981-277-949-6 As is well-known, we learn mathematics by trying to solve problems. A fascinating thing about mathematics is that even when we do not succeed in actually solving a given problem, we can still gain something by trying to solve it, for example by learning or developing new techniques.

The author presents over 150 very interesting mathemati-cal problems and their detailed solutions. The majority of the questions belong to real analysis, while some of them are taken from analytic number theory (such as those concerning the uniform distribution or a proof of the prime number theorem, which a reader accesses in an elementary way through several exercises). The book is divided into 18 chapters, each of which starts with a list of some basic knowledge in the fi eld followed by problems and their solutions. Topics include sequences and limits, infi nite series, continuous functions, differentiation, inte-gration, improper integrals, series of functions, approximation by polynomials, convex functions, the formula evaluating the Riemann zeta function at 2, functions of several variables, the uniform distribution, Rademacher functions, Legendre polyno-mials, Chebyshev polynomials, Gamma functions and the prime number theorem.

The standard of problems varies from easy, aimed at un-dergraduate students of calculus and linear algebra, to deep and complex, which could challenge experts. The text contains many useful and interesting historical comments on various signifi cant mathematical results in real analysis, and is accom-panied with corresponding references. (lp)

L. Hathout: Crimes and Mathdemeanors, A.K. Peters, Welles-ley, 2007, 196 pp., USD 14.95, ISBN 978-1-56881-260-1 As the title suggests, this book is something of a detective story with a mathematical slant. Fourteen chapters contain fourteen cases with more or less criminal plots, solved by a fourteen year old Ravi, a brilliant high school student and mathematics gen-ius who is able to reduce tangled criminal cases that baffl e po-lice and lawyers to mathematics, logic and probability, and to solve them. Amazingly, the author is a high school student him-self. He has managed to hide intriguing math problems within cleverly written criminal mysteries including murder and fraud. Readers are invited to challenge their own wits and try to solve the mysteries themselves. Each case is followed by an analysis, in which the mathematical or logical background of the crime is clearly formulated. Ravi’s brilliant solution comes next, fol-lowed by an extension to more general or more complicated mathematical questions.

Only high school mathematics (combinatorics, fi nite prob-ability, elementary inequalities, geometry, algebra, etc.) and physics is needed for solving all the problems, except one for which a bit of integration is required. Therefore the book will bring pleasure to a very broad (not necessarily mathematical) public of interested readers who like mysteries and solving co-nundrums. (lp)

P. M. Higgins: Nets, Puzzles, and Postmen. An Exploration of Mathematical Connections, Oxford University Press, Oxford, 2007, 247 pp., GBP 15.99, ISBN 978-0-19-921842-4 In this lovely and fascinating book, the author describes the world and the main functions of networks, which are all around us (from age-old examples, such as family trees, to the mod-ern phenomena of the Internet and the World Wide Web). Beginning with structures of chemical isomers, family trees, simple mathematical puzzles (for example tic-tac-toe, familiar logic games, exotic squares, Sudoku) and famous mathemati-cal problems (the bridges of Königsberg, the hand-shaking problem, the Hamilton cycle, the party problem), the author explains how networks are represented, how they work, and how they can be described, deciphered and understood. After some puzzles and games, the author introduces examples and applications of networks from natural sciences, social sciences, technology, economics, transportation sciences and genetics. Various topics are discussed (the four-colour map problem, guarding the museum, the Brouwer fi xed point theorem, the Chinese postman problem, nets as machines, automata, plan-ning routes, maximizing profi ts, the quick route, spanning networks, secret codes, RNA reconstructions, labyrinths and mazes and so on).

Understanding the book does not require a deep mathe-matical knowledge. For the reader wanting to study these top-ics from a mathematical point of view, the fi nal chapter “For Connoisseurs” can be recommended. The book will open the eyes of the reader to hidden networks, hence it can be recom-mended to people wanting to discover a remarkable new view of our world. (mbec)

A. Iosevich: A View from the Top. Analysis, Combinatorics and Number Theory, Student Mathematical Library, vol. 39, American Mathematical Society, Providence, 2007, 136 pp., USD 29, ISBN 978-0-8218-4397-0 The author has based this nice little book on a capstone course that he has taught to upper division undergraduate students, the main objective of which was to explain, in a visual way, a unity of mathematics and interactions between its fundamental areas such as analysis, algebra, number theory, topology, basic counting and probability theory. Such courses are unfortunate-ly taught rather rarely; the book will thus be a tremendous as-set and an endless source of inspiration to anyone who has a course like that in mind.

Starting with basic inequalities of Cauchy-Schwarz and Hölder, the author proceeds by pursuing their applications in geometry, then turns attention to the Besicovitch-Kakeya con-jecture on a connection of the size of a set with a number of contained line segments with different slopes, a central prob-lem in contemporary harmonic analysis, and then presents some brilliant ideas concerning basic counting (combinatorics, the binomial theorem, expected values, random walks, etc.) and probability reasoning, and fi nishes with oscillatory integrals, trigonometric sums and Fourier analysis with applications to geometry and number theory. The basic idea is to get the reader enthusiastic about mathematical research by observing the evo-lution of ideas. Most of the book requires no prerequisites (just high school mathematics) but for some chapters the reader is supposed to have certain knowledge about functions of several variables. (lp)

Recent books


M. Jarnicki, P. Pfl ug: First Steps in Several Complex Variables – Reinhardt Domains, EMS Textbooks in Mathematics, Euro-pean Mathematical Society, Zürich, 2008, 359 pp., EUR 58, ISBN 978-3-03719-049-4This book gives a nice introduction to some parts of the theory of several complex variables. It concentrates on a study of the prop-erties of a special class of domains called Reinhardt domains.

Some basic notions in the theory (domains of holomorphy, holomorphic convexity, pseudoconvexity) can be nicely intro-duced and studied in the setting of Reinhardt domains with-out too many technical details. In the fi rst chapter of the book, the authors present such an introduction to problems arising in connection to the holomorphic continuation of holomorphic functions in several complex variables. Biholomorphisms of domains and the Cartan theory in the setting of Reinhardt do-mains are treated in the second chapter. A short third chapter contains a discussion of S-domains of holomorphy for various special classes S of holomorphic functions. The last chapter is devoted to a study of holomorphically contractible families on Reinhardt domains and it includes a lot of explicit calculations in specifi c cases. Many results presented in the book are based on research by both authors. (vs)

N. L. Johnson, V. Jha, M. Biliotti: Handbook of Finite Trans-lation Planes, Pure and Applied Mathematics, vol. 289, Chap-man & Hall/CRC, Boca Raton, 2007, 861 pp., USD 99.95, ISBN 978-1-58488-605-1This book is something of a compendium of examples, proc-esses, construction techniques and models of fi nite translation planes. A translation plane is an affi ne plane that admits a group of translations acting transitively on its points. The guid-ing principle of the authors is to present an atlas of translation planes. The tremendous variety of translation planes makes any attempt at classifi cation rather diffi cult. So the authors provide a combination of descriptions and methods of construction to give an explanation of various classes of planes. Since interest in certain translation planes arises primarily from their place within certain classifi cation results, the authors also provide comprehensive sketches of the major classifi cation theorems together with outlines of their proofs. The bulk of the book is a comprehensive treatment of known examples of the planes in more than 800 pages and 105 chapters, culminating with the atlas of planes and construction processes. (jtu)

I. Kononenko, M. Kukar: Machine Learning and Data Mining. Introduction to Principles and Algorithms, Horwood Publish-ing, Chichester, 2007, 454 pp., GBP 50, ISBN 978-1-904275-21-3 Recent advances in machine learning have had a strong impact on rapid developments in related areas like data analysis, knowledge discovery and computational learning theory. This (already well-established) research discipline has found many applications in expert and business systems, data mining, databases, game play-ing, text, speech and image processing, etc. However, due to the rather interdisciplinary character of this new fi eld, many books on machine learning currently appearing are quite specialised to the considered discipline. In that sense, this book differs from sev-eral other excellent books on machine learning and data analysis, as its authors have succeeded in providing a detailed, yet com-prehensive, overview of the fi eld. The entire textbook consists of fourteen chapters and can be divided into fi ve parts.

The introductory part of the book encompasses the fi rst two chapters, which provide an historical overview of the fi eld and outline philosophical issues related to machine and human learning, intelligence and consciousness. The following four chapters deal with the basic principles of machine learning, representation of knowledge, basic search algorithms used to search hypothesis space, and attribute quality measures used to guide the search. The next two chapters set practical guidelines for preparing, cleansing and transforming the processed data. Chapters 9 – 12 are devoted to a description of the respective learning algorithms: symbolic and statistical learning, artifi cial neural networks and cluster analysis. The last two chapters in-troduce formal approaches to machine learning: the problem of identifi cation in the limit and computational learning theory.

The attached appendix reviews some theoretical concepts used in the book. Although oriented primarily towards ad-vanced undergraduate and postgraduate students, each section of the textbook is relatively self-contained and only requires a background knowledge in calculus. The discussed topics are explained in an interesting way with a lot of illustrative fi gures and supporting graphs. Each chapter is accompanied with a summary of the concepts taught. For these reasons, this book represents both a valuable teaching resource for students and a good reference source of applicable ideas for a wide audience including researchers and application specialists interested in machine learning paradigms. (im)

E. Maor: The Pythagorean Theorem. A 4,000-Year History, Princeton University Press, Princeton, 2007, 259 pp., USD 24.95, ISBN 978-0-691-12526-8 This interesting book is devoted to the Pythagorean theorem, which is the most frequently used theorem in all branches of mathematics and which is learnt in geometry at school. The au-thor shows the long route the Pythagorean theorem has taken through cultural history and he analyses its role in the develop-ment of mathematical thinking, research and teaching.

The book starts with a description of the earliest evidence of knowledge of the theorem, which was known to the Babylo-nians over 4000 years ago. It continues with an analysis of the work of the Greek mathematician Euclid, which immortalized this theorem as Proposition 47 in the fi rst book of his famous Elements. Then the book describes the role of the theorem in Greek mathematics (Archimedes, Apollonius, etc.) and its posi-tions in pure and applied mathematics, as well as its infl uence on the arts, poetry, music and culture through the Arabic world, the Middle Ages, the Renaissance and the New Age in Europe up to Einstein’s theory of relativity. The most beautiful proofs of the theorem are shown and explained. The book contains an index and many interesting photos and pictures. It can be recommended to readers who want to learn about mathematics and its history, who want to be inspired and who want to under-stand important mathematical ideas more deeply. (mbec)

E. Menzler-Trott: Logic’s Lost Genius – The Life of Gerhard Gentzen, History of Mathematics, vol. 33, American Mathemati-cal Society, Providence, 2007, 440 pp., USD 89, ISBN 978-0-8218-2550-0 This book gives (for the fi rst time in English) a detailed scientif-ic biography of Gerhard Gentzen (1909-1945), a German math-ematician who was one of the founders of modern structural

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proof theory and who is considered to be one of the greatest logicians from the fi rst half of the 20th century. In six chapters, the author describes Gentzen’s life and his scientifi c achieve-ments from his youth, through his studies, works, teaching and scientifi c activities, and his military service as well as his politi-cal ideas, until his arrest and tragic death in Prague in 1945. The author also analyses Gentzen’s conditions for scientifi c study and research in National Socialist Germany before and dur-ing World War II, his fi ghts for “German logic”, and his battles with colleagues and the political system. The book is based on a study of unknown and unpublished sources, archive materials, private letters, family documents and memoirs.

The book ends with four interesting appendices. The fi rst and second appendix (written by C. Smoryński – Gentzen and Geometry, Hilbert’s Programme) contain a short essay on Gentzen’s results in geometry and a deeper analysis of Hilbert’s programme showing relations to ideas and mathematical results of Hilbert, Brouwer, Weyl, Gödel and Gentzen. In the third ap-pendix, there are (for the fi rst time in English) three lectures by G. Gentzen (The Concept of Infi nity in Mathematics, The Concept of Infi nity and the Consistency of Mathematics, and The Current Situation in Research in the Foundation of Math-ematics), which were presented by G. Gentzen to a wide math-ematical public in Münster (1936), at the Descartes Congress in Paris (1937) and in Bad Kreuznach (1937). The fourth appen-dix (written by John von Plato) explains in detail the Gentzen mathematical program, his ordinal proof theory, his work on natural deduction and his calculus, and it gives a general survey of later developments in structural proof theory.

The book can be recommended to a wide audience; it is suitable for mathematicians, historians of science, students and teachers. (mbec)

R. E. Moore, M.J. Cloud: Computational Functional Analy-sis, second edition, Horwood Publishing, Chichester, 2007, 180 pp., GBP 27.50, ISBN 978-1-904275-24-4 This book is devoted to a brief survey of basic structures and methods of functional analysis used in computational math-ematics and numerical analysis. It is an introductory text in-tended for students at the fi rst year graduate level; despite its occasional lack of depth and precision, it provides a good op-portunity to get acquainted with the rudiments of this powerful discipline. The main emphasis is on numerical methods for op-erator equations – in particular, on analysis of approximation error in various methods for obtaining approximate solutions to equations and system of equations.

The book could be loosely divided into two parts; the former concentrates on linear operator equations, the latter on nonlin-ear operator equations. After introducing the basic framework used in functional analysis (such as linear, topological, metric, Banach and Hilbert spaces), the authors move on to linear functionals and operators and several types of convergence in Banach spaces. Special chapters are devoted to reproducing kernel Hilbert spaces and order relations in function spaces.

The fi rst part of the book fi nishes with basic elements of the Fredholm theory of compact operators on Hilbert spaces and with approximation methods for linear operator equations. The second part (concentrating on nonlinear equations) starts with interval methods for operator equations and basic fi xed point problems. After introducing Fréchet derivatives in Ba-

nach spaces and its elementary properties, their applications in Newton’s method and its variant in infi nite dimensional spaces are presented. The last chapter is devoted to a particular ex-ample of a use of the theory in a “real-world” problem; the au-thors choose a hybrid method for a free boundary problem. The topics are mostly discussed without proofs but each chapter is accompanied by a series of exercises that are designed to help students to learn how to discover mathematics for themselves. Hence the book serves as a readable introduction to functional analytical tools involved in computation. (jsp)

A. Neeman: Algebraic and Analytic Geometry, London Math-ematical Society Lecture Note Series 345, Cambridge Univer-sity Press, Cambridge, 2007, 420 pp., GBP 40, ISBN 978-0-521-70983-5 To explain basic principles of modern algebraic geometry to undergraduate students with a standard education is a very diffi cult task. Nevertheless, Amnon Neeman has succeeded in this book in showing that it is possible. The book is based on his own teaching experience and the basic idea is simple. He has chosen to present a formulation and the proof of Serre’s famous theorem on ‘géométrie algébriques and géométrie ana-lytique’ (GAGA) as the ultimate aim of the book. On the way to this goal, he introduces a lot of notions and tools of modern algebraic geometry (e.g. the spectrum of a ring, Zariski topol-ogy, schemes, algebraic and analytic coherent sheaves, localiza-tions, sheaf cohomology). The whole book shows to the reader a beautiful mixture of analysis, algebra, geometry and topology, all used together in the modern language of algebraic geometry and, at the same time, nicely illustrating its power.

The book is written in a very understandable way, with a lot of details and with many remarks and comments helping to develop intuition for the fi eld. It is an extraordinary book and it can be very useful for teachers as well as for students or non-experts from other fi elds. (vs)

R. Niedermeier: Invitation to Fixed-Parameter Algorithms, Oxford Lecture Series in Mathematics and its Applications 31, Oxford University Press, Oxford, 2006, 300 pp., GBP 55, ISBN 0-19-856607-7 Theoretical computer science as the borderline between math-ematics and computer science is a wonderland of fast evolving theories, challenging open problems and plentiful algorithmic and complexity results, all backed up by practical motivation and applications in computing. One of the recently opened and intensively studied directions is the so-called Fixed-Parameter Complexity, designed and developed in the 1990s by Downey and Fellows. Peppered with practical questions of solving prob-lems with small parameter values, problems that are computa-tionally hard in general, this is a surprisingly rich theory, where computational classes are defi ned based on precise logic con-structions. It has immediately become popular among research-ers; every major computer science conference gets a number of presentations from this area, and specialized workshops are now regularly organized. And yet, only one monograph devoted to this topic was available until recently; the book under review organically complements this previously published monograph (Downey-Fellows, Parametrized Complexity, 1999).

Downey and Fellows, the gurus of FPT (Fixed Parameter Tractability), introduce the theory by concentrating more on

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structural defi nitions and properties whilst Nidermeier comes in with numerous samples of FPT algorithms for particular problems. Moreover, a lot has happened since the Downey-Fel-lows monograph was published. The book under review is an excellent survey of new results, focused primarily on algorith-mic issues.

The book is divided into three chapters. The fi rst chapter provides necessary defi nitions accompanied by motivating ex-amples and a broad exposition of the philosophy of Fixed Pa-rameter Complexity. Don’t skip this chapter if you are a novice in this area or if you want to enjoy learning new views on it. The second chapter provides a thorough catalogue of the method-ology of fi xed parameter algorithms. This is an encyclopaedia of methods such as kernelization, depth-bounded search trees and graph decompositions, all thoroughly explained with many examples. The third chapter gives examples of hardness reduc-tions in Fixed Parameter Complexity Theory and describes the W[t] classes, with most attention paid to W[1] and W[2], which is where most of the natural problems lie.

The last section, called “Selected Case Studies”, is a Mad-ame Tussauds Museum of FPT. The difference is, you do not fi nd any wax problems here; all of them are alive and developing at the time you read about them. The exposition here is infl uenced by the famous NP-completeness bible of Garey and Johnson: “Computers and Intractability – A guide to the Theory of NP-completeness” (1979), and the problems are described as care-fully from the FPT perspective as Garey and Johnson did from the P-NP one. And not only because of this section is the book becoming as useful as Garey-Johnson’s monograph.

This is an advanced textbook intended and well-suited both for researchers and graduate students in theoretical computer science. You will enjoy reading it whether you want to do re-search in the area or just want to learn about it. And while mer-rily intending the latter, you are most likely going to end up doing the former when you fi nish reading it. (jkrat)

V. V. Prasolov: Elements of Homology Theory, Graduate Stud-ies in Mathematics, vol. 81, American Mathematical Society, Providence, 2007, 418 pp., USD 69, ISBN 978-0-8218-3812-9 This book is a translation of the Russian original (published in 2005). It is a continuation of the author’s earlier book “Ele-ments of Combinatorial and Differential Topology”. It is con-venient to have the companion volume at hand because the book refers to it for basic defi nitions and concepts. In the fi rst half of the book, the author builds the theory of simplicial ho-mology and cohomology and describes some of its applications (most of the space being devoted to characteristic classes in the simplicial setting). It is followed by a description of singular homology. The penultimate chapter defi nes Čech cohomology and de Rham cohomology and it proves the de Rham theorem and its simplicial analogue. The last chapter “Miscellany” con-tains material on invariants of links, embeddings and immer-sions of manifolds, and a section on cohomology of Lie Groups and H-Spaces.

The book contains 136 problems (mostly in the chapters on simplicial homology and cohomology and their applications). For the majority of problems, there are solutions or hints in a 37-page appendix. Together with its companion volume, the book can be recommended as a basis for an introductory course in algebraic topology. (dsm)

G. Royer: An Initiation to Logarithmic Sobolev Inequalities, SMF/AMS Texts and Monographs, vol. 14, American Math-ematical Society, Providence, 2007, 119 pp., USD 39, ISBN 978-0-8218-4401-4 Classical Sobolev inequalities constitute an extremely important part of functional analysis with a surprisingly wide fi eld of appli-cations. The study of certain specifi c topics such as quantum fi elds and hypercontractivity semigroups requires an extension of a classical Sobolev inequality to the setting of infi nitely many vari-ables. This is a diffi cult problem because the Lebesgue measure in infi nitely many variables is meaningless. A solution was dis-covered by Leonard Gross in his famous pioneering paper from 1975, where he replaced the Lebesgue measure with the Gauss measure. In the end, the power integrability gain known from classical inequalites is replaced by a more delicate logarithmic growth – hence the name. Gross’ logarithmic Sobolev inequali-ties have found, again, an impressive range of applications.

The book under review provides an introduction to loga-rithmic Sobolev inequalities and to one of its specifi c applica-tions in the fi eld of mathematical statistical physics, more pre-cisely to the example concerning real spin models with weak interactions on a lattice. A proof of the uniqueness of the Gibbs measure is given based on the exponential stabilization of the stochastic evolution of an infi nite-dimensional diffusion proc-ess, a generalization of the Ising model. The author begins with the background material on self-adjoint operators and semi-groups, then proceeds to logarithmic Sobolev inequalities with applications to Kolmogorov diffusion processes, and fi nishes with Gibbs measures and the Ising models. The text is comple-mented with exercises and appendices that extend the material to related areas such as Markov chains. (lp)

A. A. Samarskii, P. N. Vabishchevich: Numerical Methods for Solving Inverse Problems of Mathematical Physics, Inverse and Ill-Posed Problems Series, Walter de Gruyter, Berlin, 2007, 438 pp., EUR 148, ISBN 978-3-11-019666-5In this monograph, the authors consider the main classes of inverse problems in mathematical physics and their numerical treatment. They include in the book many numerical illustra-tions and codes for their realization. They give a rather com-plete treatment of basic diffi culties for approximate solutions of inverse problems. They use minimal mathematical apparatus (in particular, basic properties of operators in fi nite-dimension-al spaces). Russian mathematicians contributed in a substantial way to solutions of theoretical and practical questions studied in inverse problems in mathematical physics (pioneering work in this direction was done by Andrei Nikolaevich Tikhonov).

The main topics treated in the book are boundary value problems for ordinary differential equations and elliptic and parabolic partial differential equations, methods of solutions for ill-posed problems and evolutionary inverse problems. The book is intended for graduate students and scientists interested in applied mathematics, computational mathematics and math-ematical modelling. (knaj)

C. G. Small: Functional Equations and How to Solve Them, Problem Books in Mathematics, Springer, Berlin, 2007, 129 pp., EUR 32.95, ISBN 978-0-387-34539-0This book is devoted to functional equations of a special type, namely to those appearing in competitions like the Interna-

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tional Mathematical Olympiad for high school students or in the William Lowell Putnam Competition for undergraduates. Its aim is to present methods of solving functional equations and related problems and to provide basic information on its history. The intention limits the generality of the treatment; more complicated cases are omitted to keep the exposition understandable for secondary school students. To give a feel-ing of how it is done I will describe the content of the fi rst two chapters. In the introductory part, contributions by Nicole Oresme, Gregory of Saint Vincent, Cauchy, d’Alembert, Bab-bage and Ramanujan are presented together with some facts from their lives. Also, a simultaneous solution of two equa-tions is found and then the terminology is fi xed. The chapter ends with some simple problems testing understanding of the subject.

The second chapter starts with the Cauchy equation for additive functions and with Jensen’s equation. Some gener-alizations like Pexider’s and Vincze’s equations are treated. Cauchy’s inequality for subadditive functions is studied as well as the Euler and d’Alembert equations. The author is able in such a way to show important types of equations and ways/tricks necessary for dealing with problems containing them. The book contains many solved examples and problems at the end of each chapter. More than 25 pages at the end of the book offer hints and briefl y formulated solutions of those problems. Also a short appendix on Hammel basis is included. The book has 130 pages, 5 chapters and an appendix, a Hints/Solutions section, a short bibliography and an index. It has a nice and clear exposition and is therefore a very readable book, accessi-ble without special or highly advanced mathematics. The book will be valuable for instructors working with young gifted stu-dents in problem solving seminars. (jive)

L. Smith: Chaos – A Very Short Introduction, Oxford Uni-versity Press, Oxford, 2007, 180 pp., GBP 6.99, ISBN 978-0-19-285378-3 Chaos is everywhere. It is a behaviour that looks random but is not random. It is also an important area of contemporary math-ematical research, studying situations where very small changes at the beginning of a certain process cause huge errors at the end. This is not only about weather forecasting or fi ve-year economy planning that has been used by communist systems. The idea of a small mistake having enormous consequences after some time is intuitive, yet it leads to new ways of under-standing physical sciences. Chaos is a scientifi c discipline, which has recently been developing rapidly with interesting facts be-ing discovered. It is also littered with pervasive myths such as the one about non-predictability of chaotic systems, which calls for explanation.

Everyone interested in such an explanation as well as in questions, like whether the fl ap of a butterfl y’s wing can affect a hurricane on the other side of the world, should read this book. The author provides an excellent introductory explanation to the fascinating topic of chaos, on an accessible level and in a highly entertaining way. (lp)

T. Timmermann: An invitation to quantum groups and dual-ity. From Hopf algebras to multiplicative unitaries and be-yond, EMS Textbooks in mathematics, EMS Publishing House, Zürich, 2008, 407 pp., EUR 58, ISBN 978-3-03719-043-2

The Pontrjagin duality for locally compact Abelian groups is the best setting for a generalization of classical harmonic analy-sis and Fourier transforms. Dual objects to compact Abelian groups are discrete groups. To have a suitable analogue for non-commutative locally compact groups, it is necessary to fi nd a larger category containing both groups and their duals. Such a broader scheme was recently developed under the infl uence of new ideas coming from physics (quantum groups). It gives an interpretation of quantum groups in the setting of C*-algebras and von Neumann algebras. The compact case (developed by Woronowicz) is included as a special case.

This book is devoted to a description of this theory. It has three parts. The fi rst part treats quantum groups (and their duality) in a purely algebraic setting. It contains a description of the Van Daele duality of algebraic quantum groups (giv-ing a model for further generalizations) and a discussion of the Woronowicz compact quantum groups. In the second part, quantum groups are treated in the setting of C*-algebras and von Neumann algebras. In particular, it contains a presentation of the Woronowicz of C*-algebraic compact quantum groups and a study of multiplicative unitaries. The last part contains several topics. One is the cross product construction and the duality theorem (due to Baaj and Skandalis), the other is pseu-do-multiplicative unitaries on Hilbert spaces. The last chapter contains the author’s results on pseudo-multiplicative unitaries on C*-modules.

The book is nicely written and very well organized. It offers an excellent possibility for students and non-experts to learn this elegant new part of mathematics. (vs)

K. Zhu: Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics, vol. 226, Springer, Berlin, 2005, 268 pp., EUR 69,95, ISBN 0-387-22036-4 This book is devoted to the study of various spaces of holo-morphic functions on the unit ball. The main tool used for their study is an explicit form of the Bergman and Cauchy-Szëgo kernels. There is a lot of different spaces of that sort and the author has chosen some of them for a detailed study. There is always one chapter of the book devoted to one type of func-tion spaces. The spaces treated in the book are Bergman spaces, Bloch space, Hardy spaces, spaces of functions with bounded mean oscillation (BMO), Besov spaces and Lipschitz spaces. The author discusses the various characterizations, atomic de-compositions, interpolations, and duality properties of each type of function space in turn. To read the book, knowledge of standard complex function theory is expected but the theory of several complex variables is not a prerequisite. Results pre-sented are standard but their proofs are often new. (vs)

Hemodynamical FlowsModeling, Analysis and Simulation

Galdi, G.P., University of Pittsburgh, PA, USA / Rannacher, R., Universität Heidelberg, Germany / Robertson, A.M., University of Pittsburgh, PA, USA / Turek, S.,Universität Dortmund, Germany

2008. XII, 501 p. SoftcoverEUR 49.90 / CHF 65.00ISBN 978-3-7643-7805-9OWS — Oberwolfach Seminars, Vol. 37

This book surveys results on the physical and mathematical modeling as well as the numerical simulation of hemodynamical ows, i.e., of uid and structural mechanical processes occurring in the human blood circuit. The topics treated are continuum mechanical description, choice of suitable liquid and wall models, mathematical analysis of coupled models, numerical methods for ow simulation, parameter identi cation and model calibration, uid-solid interaction, mathematical analysis of piping systems, particle transport in channels and pipes, arti cial boundary conditions, and many more. Hemodynamics is an area of active current research, and this book provides an entry into the eld for graduate students and researchers. It has grown out of a series of lectures given by the authors at the Oberwolfach Research Institute in November, 2005.

Vanishing and Finiteness Results in Geometric AnalysisA Generalization of the Bochner Technique

Pigola, S., Università dell‘Insubria, Como, Italia / Rigoli, M., Università di Milano, Italia / Setti, A.G., Università dell‘Insubria, Como, Italia

2008. XIV, 282 p. HardcoverEUR 49.90 / CHF 89.90ISBN 978-3-7643-8641-2PM — Progress in Mathematics, Vol. 266

This book presents very recent results involving an extensive use of analytical tools in the study of geometrical and topological properties of complete Riemannian manifolds. It analyzes in detail an extension of the Bochner technique to the non compact setting, yielding conditions which ensure that solutions of geometrically signi cant differential equations either are trivial (vanishing results) or give rise to nite dimensional vector spaces ( niteness results). The book develops a range of methods from spectral theory and qualitative properties of solutions of PDEs to comparison theorems in Riemannian geometry and potential theory. All needed tools are described in detail, often with an original approach. Some of the applications presented concern the topology at in nity of submanifolds, Lp cohomology, metric rigidity of manifolds with positive spectrum, and structure theorems for Kähler manifolds. The book is essentially self-contained and supplies in an original presentation the necessary background material not easily available in book form.

Studies in Universal LogicEditorial Board Members: H. Andréka (Hungarian Academy of Sciences, Hungary), M. Burgin(University of California, USA), R. Diaconescu (Romanian Academy, Romania), J. M. Font (University of Barcelona, Spain), A. Herzig (CNRS, France), A. Koslow (City University of New York, USA), J.-L. Lee (NationalChung-Cheng University, Taiwan), L. Maksimova (Russian Academy of Sciences, Russia), G. Malinowski (University of Lodz, Poland), D. Sarenac(Stanford University, USA), P. Schröder-Heister (Universität Tübingen, Germany), V. Vasyukov (Academy of Sciences Moscow, Russia)

This series is devoted to the universal approach to logic and the development of a general theory of logics. It covers topics such as global set-ups for fundamental theorems of logic and frameworks for the study of logics, in particular logical matrices, Kripke structures, combination of logics, categorical logic, abstract proof theory, consequence operators, and algebraic logic. It includes also books with historical and philosophical discussions about the nature and scope of logic. Three types of books will appear in the series: graduate textbooks, research monographs, and volumes with contributed papers.

Completeness Theory for Propositional LogicsPogorzelski, W.A., University of Bialystok, Poland / Wojtylak, P., Silesian University Katowice, Poland2008. VIII, 178 p. SoftcoverEUR 49.90 / CHF 85.00 / ISBN 978-3-7643-8517-0

Institution-independent Model TheoryDiaconescu, R., Simion Stoilow Romanian Academy, Romania2008. Approx. 390 p. SoftcoverEUR 69.90 / CHF 125.00 / ISBN 978-3-7643-8707-5

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Lectures on Advances in CombinatoricsR. Ahlswede, University of Bielefeld, Germany; V. Blinovsky, Russian Academy of Sciences, Moscow, Russua

The book contains a complete presentation of some highly sophisticated proofs following new methods of pushing and pulling appearing mostly for the first time in a book. New in a book are also several diametric theorems for sequence spaces, bounds for list codes, and even the concepts of higher level and dimension constrained extremal problems, and splitting antichains. Furthermore for the first time some number-theoretical and combinatorial extremal problems are treated in parallel and thus connections are made more transparent.

2008. Approx. 310 p. (Universitext) SoftcoverISBN 978-3-540-78601-6 € 39,95 | £30.50

An Introduction to Mathematical CryptographyJ. Hoffstein, J. Pipher, J. Silverman, Brown Univer-sity, Providence, RI, USA

An Introduction to Mathematical Cryptography provides an introduction to public key cryptog-raphy and underlying mathematics that is required for the subject. Each of the eight chapters expands on a specific area of mathematical cryptography and provides an extensive list of exercises.It is a suitable text for advanced students in pure and applied mathematics and computer science, or the book may be used as a self-study. This book also provides a self-contained treatment of mathematical cryptography for the reader with limited mathematical background.

2008. XVI, 524 p. 29 illus. (Undergraduate Texts in Mathematics) HardcoverISBN 978-0-387-77993-5 € 34,95 | £26.50

Analysis by Its HistoryG. Wanner, E. Hairer, University of Geneva, Switzerland

This book presents first-year calculus roughly in the order in which it first was discovered. The first two chapters show how the ancient calculations of prac-tical problems led to infinite series, differential and integral calculus and to differential equations. The establishment of mathematical rigour for these subjects in the 19th century for one and several variables is treated in chapters III and IV. The text is complemented by a large number of examples, calculations and mathematical pictures.

2008. Approx. 390 p. 192 illus. SoftcoverISBN 978-0-387-77031-4 € 32,95 | £25.50

Generalized CurvaturesJ. Morvan, Université Claude Bernard Lyon 1, Villeurbanne, France

Following a historical and didactic approach, the book introduces the mathemat-

ical background of the subject, beginning with curves and surfaces, going on with convex subsets, smooth submanifolds, subsets of positive reach, polyhedra and triangulations, and ending with surface reconstruction. It can serve as a textbook to any mathematician or computer scientist, engineer or researcher who is interested in the theory of curvature measures.

2008. IX, 266 p. 107 illus., 36 in color. (Geometry and Computing, Volume 2) HardcoverISBN 978-3-540-73791-9 € 79,95 | £61.50

Braid GroupsC. Kassel, Université Louis Pasteur - CNRS, Strasbourg, France; V. Turaev, Indiana University, Indiana, US

In this well-written presentation, motivated by numerous examples and problems, the authors introduce the basic theory of braid groups, highlighting several definitions that show their equivalence; this is followed by a treatment of the relationship between braids, knots and links. Important results then treat the linearity and orderability of the subject. Relevant additional material is included in five large appendices. Braid Groups will serve graduate students and a number of mathematicians coming from diverse disciplines.

2008. XII, 340 p. 60 illus. (Graduate Texts in Mathematics, Volume 247) HardcoverISBN 978-0-387-33841-5 € 42,95 | £32.50

Boundary Integral EquationsG. C. Hsiao, University of Delaware, Newark, DE, USA; W. L. Wendland, University of Stuttgart, Germany

This book is devoted to the basic mathematical properties of solutions to boundary integral equations and presents a systematic approach to the variational methods for the boundary integral equations arising in elasticity, fluid mechanics, and acoustic scattering theory. It may also serve as the mathematical foundation of the boundary element methods.

2008. Approx. 635 p. (Applied Mathematical Sciences, Volume 164) HardcoverISBN 978-3-540-15284-2 € 79,95 | £60.00

Catalan’s ConjectureR. Schoof, Università di Roma, Italia

Eugène Charles Catalan made his famous conjecture – that 8 and 9 are the only two consecu-

tive perfect powers of natural numbers – in 1844 in a letter to the editor of Crelle’s mathematical journal. One hundred and fifty-eight years later, Preda Mihailescu proved it.Catalan’s Conjecture presents this spectacular result in a way that is accessible to the advanced undergraduate. The author dissects both Mihailescu’s proof and the earlier work it made use of, taking great care to select streamlined and transparent versions of the arguments and to keep the text self-contained. Only in the proof of Thaine’s theorem is a little class field theory used; it is hoped that this application will motivate the interested reader to study the theory further.

2008. Approx. 150 p. 10 illus. (Universitext) SoftcoverISBN 978-1-84800-184-8 € 39,95 | £27.00

Subdivision SurfacesJ. Peters, University of Florida, Gainesville, FL, USA; U. Reif, TU Darmstadt, Germany

This book summarizes the current knowledge on the

subject. It contains both meanwhile classical results as well as brand-new, unpublished material, such as a new framework for constructing C

2-algorithms. The focus of the book

is on the development of a comprehensive mathematical theory, and less on algorithmic aspects. It is intended to serve researchers and engineers - both new to the beauty of the subject - as well as experts, academic teachers and graduate students or, in short, anybody who is interested in the foundations of this flourishing branch of applied geometry.

2008. XVI, 204 p. 52 illus., 8 in color. (Geometry and Computing, Volume 3) HardcoverISBN 978-3-540-76405-2 € 49,95 | £38.50

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OF THE EUROPEAN MATHEMATICAL SOCIETY · Analysis and Stochastics of Growth Processes and Interface Models Peter Mörters, Roger Moser, Mathew Penrose, Hartmut Schwetlick and Johannes